Jan Robert Selman UCRL-20 557 o.Q
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MEASUREMENT AND INTERPRETATION OF LIMITING CURRENTS
Jan Robert Selman (PhD, Thesis) June 1971
AEC Contract No0 W-7405-eng-48
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Table of Contents Abstract . Introduction
Chapter 1. Applicability of the limiting current method Part The limiting current method
I.I. The limiting current ...... 3 1.2. Measurement of mass transfer rates ...... 5 1.3. Model reactions ...... 7 * 1.4. Historical ...... 11 Part Review-of applications 1.5. Simple laminar flows ...... 14 1.6. Laminar flow with complications ...... 25 1.7. Turbulent flows and mixed convcctjcrt ...... 40 1.8; Restricted cells with agitation ...... 52 1.9. Cells with gas-evolution ...... 55 1.10. Particulate flow systems ...... 58 1.11. Electrochemiluminescence ...... 61 1.12 Conclusions ...... 61 Chapter 2. Validity of the limiting current method
Part Basic concepts of electrochernical mass transfer 2.1. Migration and diffusion ...... 67 2.2. Polarization ...... 77 2.3. Current and potential distribution ...... 91 Part Conditions for validity 2.4. Definition of the ii nitin, current pIteau 105 2.5. Conç1ition' ror valid mcxcu1epnt 105 2.6. (oditiç for valjc1 1ntcF1ctat1on 10u -11-
Chapter 3. Rotating disk integral diffusivities of Cu ion in CuSO4 - H2SO4 -H20
3 .1.. Introduction ...... 108
3 .2. Experimental ...... 110
3.3. Results • ...... 118
3.4. Discussion: migration effect ...... 138
3.5. Discussion: deposit roughness ...... 147 3.6. Conclusions ...... 157
Chapter 4. Unsteady state effects in electrochemical flow cells Part I. Review
4.1. Introduction ...... 161
4 .2. Free conyection . ...... 162
4 .3. Forced convection ...... 169 4.4. Interaction with potential. distribution at
elongated electrodes . • ...... 172 Part II. Unsteady state at a rotating disk electrode -
4 .5. Experimental ...... 177
4 .6. Results ...... 179
4 .7. Discussion ...... 194
4.8. Conclusions ...... 208
Conclusions and Recommendations ...... 210
Acknowledgments ...... 2]3
• Appendices . . • •
• A. Program for calculation of a theoretical limiting
currcnt curve -. . ...... ... . 214 -ill-
Flux at a rotating disk electrode after a concentration
step at the surface, in the limitof high Schmidt numbers . . 216
Degree of dissociation of HS0 ion in the system CuSO 4
-H2SO4-H20 237 Migration at a rotating disk electrode in acidified solutions
of CuSO4 characterized by the bisulfate dissociation
parameter I/K = 5 ...... 250
Theoretical solutions for transient convective diffusion . . 256 Transient diffusion following linear potential decrease at a
redox electrode ...... 261 C. Transient diffusion following linear' potential decrease for
a deposition reaction, taking electrode kinetics
into account ...... 270
Nomenclature ...... 288
References...... 295 MEASUREMENT AND INTERPRETATION OF LIMITING CURRENTS
Jan Robert Selman
Inorganic Materials Research Division, Lawrence Radiation Laboratory Department of Chemical Engineering University of California, Berkeley, California
June 1971
ABSTRACT
Applications of the limiting current method for measuring mass transfer rates are reviewed. To date almost 100 publications using this method have appeared, with recent applications in the area of stirred tanks, parti- culate systems, mixed convection and turbulence.
The conditions for valid measurement and interpretation of limiting currents are discussed. An ideal limiting current curve is constructed for the copper deposition model reaction. The effect of migration due to the electric field is computed for this reaction, taking into account that the bisulfate ion is not completely dissociated. The use of integral dif - fusivities, which reflect the effect of migration as well as that of variable physical properties, is required in the correlation of mass transfer coefficients calculated from limiting currents.
Rotating disk integral diffusivities are reported for cupric ion in acidified solutions of cupric sulfate In solutions containing less than 0.1 molar cupric sulfate the product of viscosity and integral diffusivity is independent of the ionic strength of the solution. In more concentrated solutions the effect of surface roughness developing near the limiting current is discernible in high dffusivitv values. Some semi-quantitative
COflSlderdtlons on the developncnt-ard extent of this ioughness effect are given -vi -
Unsteady-state effects in limiting current measurements are reviewed from the viewpoint of nonstationary convective diffusion. The complicating effect of the presence of an electric field is discussed, and it is shown that apparent, nonsimultaneous, sectional limiting currents can be expected to occur at elongated electrodes. Unsteady-state limiting currents at a rotating disk electrode are investigated experimentally, using the fern- cyartide-ferrocyanide redox system. The minimum time to reach a steady-state limiting current by linear current increase is an order of magnitude larger than that necessary if a linear potential scan is applied. Apparent limiting currents due to fast current increase or potential scan rates can be satis- factorily interpreted in terms of a pure-diffusion model.
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I
INTRODUCTION : STRUCTURE OF THE THESIS
Electrochemical measurements of mass transfer rates by the limiting
current technique have been employed in recent decades with increasing
frequency. The present work aims to contribute to the methodology of limit-
ing current measurements by a unifying review of applications and of con-
ditions for validity, as well as by a detailed study of some critically important aspects.
In the first Chapter the basic features of the method are described,
its advantages and limitations.. The varIety of mass transfer situations
towhich it has been applied, is reviewed. . . .
Chapter 2 gives a synopsis of those concepts of electrochemical
mass transfer theory which are necessary 'for an analysis of the limiting
current'.conditjon. The requirements for an. effective meaurement and a
valid interpretation of the results, are reviewed.
Three practical aspects of the limiting current technique are singled out for a detailed investigation: diffusivity values appropriate
to the method, unsteady-state mass transfer during the approach of the
limiting current, and the effect of current distribution below and at
the limiting current, due to the presence of an electric field.
Chapter 3 treats the problem of appropriate diffusivities to be used in limiting current studies, in particular for the copper deposition
reaction frequently used in free-convection studies It is shown that migration and surface roughness formed near the limiting current both have a considerable effect on the integral diffusivities measured at a rotating disk electrode The conditions under which the surface roughnes'- effect is minimal are investigated.
Chapter 4 deals with the unsteady state due to mass transfer.
Available information concerning transients in convective diffusion is
• reviewed. From a comparison with experimental limiting current curves
at planar, sectioned electrodes it can be inferred that transient ef-
fects are more persistent than expected. This is shown to be related
to the current distribution below the limiting current. • • • The second part of Chapter 4 is devoted to astudyof unsteady-
state mass transfer at a rotating disk electrode, using a redox reaction where such current distribution effects are not important. The apparent
limiting currents observed, in fast current ramps or potential scans are • . explained in terms of a pure-diffusion model. •
/ • • • 3
Chapter 1 APPLICABILITY OF THE LIMITING CURRENT METHOD Part 1. The limiting current method
1.1. The limiting current.
• The term "limiting current density" is used to describe the maximum rate at which a particular electrode reaction can proceed in the steady state, with 100% current efficiency. This rate is determined by the composition and transport properties of the electrolytic solution on the one hand, and on the other by the hydrodynamics of the electrochemical cell In a given situation the limiting current at an electrode can only be exceeded by increasing the cell pOtential until another consecutive, electrode reaction starts occurring at finite rate
The limiting current may be due to a slow,rate-determining step in the electrode reaction (kinetic limiting current). However, in the majority of cases where a limiting current is encountered it is caused by the slowness of transport of ionic or uncharged species through the solution The ions (or molecules) move toward the electrode where they are consumed in the electrode reaction, or a reaction coupled with it
Electrode reactions, by their nature, are surface reactions and cause local changes in composition A thin diffusion layer, impoverished in the reacting species, develops at the electrode surfaces The ions or mole- cules move across the diffusion layer by diffusion down concentration gradients. Ions move also under the influence of the electric field, i e , they migrate
Limiting currents are usually associated with cathodic reactions (e g in metal deposition), although anodic reactions are by no means excluded
Whenever the supply of a dissolved species from the solution to the elec- trode surface becomes a rate limiting factor, limiting Lurrent phenomena 4
may be observed. Agar.and Bowden 122 showed that diffusion of H20 limits
0 2 evolution in fused NaOH. Hoar and Rothwell 28 observed anodic limiting currents in electropolishing of copper, due in part to a dissolution
diffusion layer, and in part to a solid oxide film.
In most cases the diffusion layer is not a stagnant layer, as
presented in the early Nernst-Whitman model. Where no forced convection
is maintained free convection usually sets in due to the density gradient
developing in the diffusion layer. The convective velocity, although it
vanishes at the wall, enhances the diffusion toward the electrode. It
steepens the concentration gradient of the reacting species at the elec-
trode and therefore increases the maximum current obtainable.
Equivalently this is often expressed as a decrease of the Nernst
layer thickness. In the stagnant layer model referred to above the con-
centration profile of the reacting ion is linear, with the thickness
chosen so as to give the actual concentration gradient at the electrode.
This fictitious 6 is always smaller than the real diffusion layer thick- ness, but better defined since the actual concentration profile tapers off very gradually to the bulk value.
clearly the maximum gradient for a certain rate of stirring, i.e., a given hydrodynamic condition near the electrode, is that gradient where- by the concentration at the electrode is reduced to zero, or virtually zero. This situation corresponds, in an electrochemical cell, to the limiting current condition. S
1 2 Measurement of mass transfer rates
If the rate ofthe reaction is not restricted by the kinetics of
the reaction mechanism, the reaction is called "reversible and its
rate is transport-controlled The current density, i, is proportional to
the reacting ion (or molecule) flux, N., byFaraday's law:
s.i = -nFN. (1.1) 1 - -1
Here.s is the number of molecules of speciesi participating in the
transfer of n electrons to or from the electrode.
Therefore, if the driving force of the mass transfer process can be expressed in a practical form, e.g., as a concentration gradient of the reacting species, or as a concentration difference betw.een electrode and bulk solution, one can relate the mass transfer rate in a general way to the concentration driving force. If the concentration gradient is known, an effective diffusivity can be defined
N = _Deff Vc (1.2)
More often, only the concentration difference electrode/bulk is known, and in such cases the mass-transfer coefficient can be calculated:
N = k(clO_clb) (1.3)
In the limiting current method of measuring mass transfer coefficients, the reacting ion concentration at the electrode is made vanishingly small by applying a large potential In that case only the current density and the bulk concentration need to be known 6
If the concentration at the electrode is not zero, it can still be measured by optical techniques, in particular by interferometric methods.
Such measurements, however, are relatively simple only in dilute solu 115 tions, because refraction occurs in addition to interference. More- over, they are restricted to solutions where only the reacting species concentration varies, i e , to solutions of a single salt If the solu- tion contains two electrolytes with dissimilar concentration profiles in the diffusion layer, a second, independent, measurement is needed to establish the reactant concentration at the electrode.
In practice such multicomponent electrolytic solutions are used much more frequentlythan those of singlesalts. In binary solutions the current exceeds the expected current due to. convective diffusion by a factor of approximately two. This excess current is due to migration of the reacting . ion in the electric field. It can be suppres,sedby increasing the conductivity of the solution, thereby lowering the elec- tric field strength. Thus, in mass transfer measurements, a large amount of "supporting" or "inert" electrolyte is usually added to the solution.
The complications associated with the presence of non-reacting ions in the diffusion layer have been discussed extensively elsewhere1131114
They. are of special importance in free convection, where the driving force, i.e., a density difference, is affected by the accumulation or deficit of non-reacting ions at the electrode. In both forced and free convection it is important to know the ion flux contributed by migration, which is never completely absent. Chapter 2 deals with this problem in quantitative terms. 7
Since current measurements can be made very exactly, the limiting
current technique is a convenient and precise method of measuring mass
transfer rates. The limiting current makes itself knownas a plateau
or inflection point in the plot of current versus electrode overpotential
(Figure 3.6). In principle the accuracy of mass transfermeasurements is
limited only by the precision with which this limiting current plateau,
or point, can be read. Further, the electrode area, current distribution
and reactant bulk concentration have to be known accurately.
1.3. Model reactions.
The electrochemjcal reactions most often used for the study of
limiting currents are listed in Table 1.1. Oxygen reduction was used • 87 . in early experiments with a rotating disk. Later many investigators made use of copper deposition from acidified solution. This reaction
has the disadvantage of changing the cathode surface, especially near
the limiting current where the deposit becomes rough (see section 3 5 )
For forced convection studies copper deposition has been largely
supplanted by the reduction of ferricyanide at a nickel or platinum
surface. Usually, an equimolar mixture of fern- and ferrocyanide is
used. The oxidation of ferrocyanide at the anode compensates, at least
in part, for the cathodic depletion of ferricyanide, if anolyte and catholyte are not separated by a diaphragm The ferrocyanide oxidation reaction has also been used occasionally as the controlling reaction, but it has the disadvantage of a shorter limiting current plateau In addition, oxygen evolution passivates a nickel anode (Both electrodes 8
Table Li. Model reactions in electrochemical mass transfer studies. Ilk
E°(v) electrode supporting reference
metal electrolyte -
CAThODIC many 1 Fe(CN) + e -* Fe(CN) +0 360 Ni,Pt 6 6
NaOH 2,87 2 02 + 2H20 + 4e - 40H +0 401 Pt,Ag
3 1 3 + 2e -* 31 +0 536 Pt KI 59,88
fosfate 4 +2H + 2e - +0 699 Ag 2 0 OOH buffer
s z2n + 2e Zn -0 763 Zn(Hg) KOH
6 Cd 2 + 2e - Cd -0 352 Cd(Hg) none 4 2+ H SO many 7 Cu + 2e Cu +0. 337 Cu 2 4 none 7,9,23
8 Ag + e - Ag +0 799 Ag HC10 4 5
ANODIC
9 CuCl -* CuCl + e +0 153 Pt CaCl 15 4 4 2 many 10 Fe(CN) -* Fe(CN) + e +0 360 Ni,Pt
4 9
are susceptible to CN poisoning. The correct procedure to activate
nickel electrodes before use has been described by Eisènberg c.s. 6 )
For studies of free convection mass transfer, or combinations of
forced and free convection, the copper deposition reaction remains the
preferred reaction, in spite of the precautions necessary to prevent
excessive surface roughening near the limiting current The advantage that copper deposition offers in such studies is a considerably larger driving density difference, due to (1) its higher solubi.lity; (2) the
larger densification (see Table 1.2).
In the case of the redox reaction the opposite effects of reactant depletion and product accumulation canceleach other to a large extent.
The densification due to ferrocyanide is larger than that due to fern- cyanide (each larger than that for CuSO4 ), so that the redox reaction makes the catholyte denser and the anolyte less dense
The actual densification is also influenced by the effect of migration. As shown in the third line of Table 1.2, the relative densifi- cation in acidified cupric sulfate is less than in binary cupric sulfate solution. In the case of the supported redox reaction the migration effect makes the density difference larger than expected from over-all reaction stoichiometry.
None of the other reactions listed in Table 1.1 have attained the popularity of ferricyanide reduction and copperdeposition. The oxygen and iodine reduction reactions satisfy the requirements of a reasonably long plateau (> 250 my, depending on the supporting electrolyte pH) and do not change the surface They are however restricted to forced convec- tion situations with large flow rates, since the bulk concentrations are 10
Table 1.2. Maximum densification in cupric sulfate and alkaline equimolar fern-and ferrocyanide solutions, at 25 ° C.
CuSO4 CuSO4 equimolar ferri/ferrocyanide +2.OM NaOH
+l.5M HS02 4
cathodic anodic
LcR(M) 1.4 0.9 0.2 0.2 cLR(M1) 0.140 0 140 0.167 0 226 tsp/p 0.140 0 125 -0 0706 0.0832 p(g/ml) 1.214 1.217 1.50
(cp) 1.824 1.914 1.647 v(cst) 1.502 1.573 1.432
Ap(g/ml) 0.170 0.137 -0 0812 +0 0957
Values of aR and Ap/p taken from Ref. 114
1. 11
of the order of 10 3M at most. Reducible impurities and gases (e.g.
02 if iodine is the reactant) have to be rigorously excluded In the case of oxygen reduction, maintaining a known oxygen concentration in the bulk.is not a trivial problem because of the tendency to super- saturation, and the extreme sensitivity of the solubility to the salt concentration in solution.
The silver and zinc deposition reactions have only occasionally been used. Dendritic deposits form very easily unless organic inhibitants are added,. which complicates the electrode kinetics to the point of obscuring the limiting current
In the present study only ferricyanide reduction and copperdeposition were employed.
144. Histo±ica1.
The dependence of current density on the rate of stirring was first 116 117 clearly established by Nernst and Brunner (1904). They interpreted this by means of the stagnant layer concept first used by Noyes and Whitney.
The thickness of this layer ("Nernst diffusion layer thickness") was correlated simply with the speed of the stirring impeller or rotating electrode tip
The lack of hydrodynamic definition was recognized by Eucken, 99 who considered convective diffusion transverse to a parallel flow and obtained an expression analogous to the Lvque equation of heat transfer.
Experiments with Couette flow between a rotating cylinder and a station- ary outer cylinder did not confirm his predictions At very low rotation rates, where such a flow is stable, it does not contribute to the diffusion process since there is no velocity component in the radial 12
direction At higher rotation rates secondary flow patterns form (Taylor
vortices) and finally the flow becomes turbulent Neither flow regime satisfies the conuitions of the Lvque equation
However, flow generated by a cylinder rotating at high speed was 118 119 subsequently used by others, in particular King and co-workers ' to demonstrate that dissolution proLesses are diffusion limited, and electro-
chemical corrosion processes as well The dependence of the mass-transfer coefficient on the 0.7 power of rotation rate and diffusivity of the
dissolving species was established (cf. Table 1.3, eq. 31).
Of considerable importance for the conceptual development of limit-
ing current measurement was the work of Agar and Bowden 122 on the current- overpotential relationship at nickel electrodes in fused sodium hydroxyde, where water transport is the limiting step
2OH+-O2 +H2O+e
A more detailed review of this, and other, early work in ionic mass
transfer is given by Tobias, Wilke and Eisenberg 100
The full development of electrochemicai techniques of mass transfer
measurement had to await the formulation of a quantitative insight into
the role of convective diffusion in electrode processes. This was first
successfully provided by the groundbreaking work of Levich on the rotating
disk electrode,9 5 ' 106 so that by 1947 the conditions were fuli11ed for
a matching of theory and practice. 87 Agar 107 pointed out that electro- chemical mass transfer correlations should be closely similar to those known for heat transfer The first test of the theory of convective diffusion in free convection was made by Wagner. 1 From 1951 on limiting 13
current measurements have been used with increasing frequency and Lonfidence
to establish mass transfer •rates.at reactive surfaces in flow situations
where they cannot be predicted on theoretical grounds, or where the theory needs confirmation.
Only a few reviews have appeared in which the application of the
method is discussed from a chemical engineering viewpoint In the review
already mentioned Tobias, Wilke and Eisenberg 100 examined the available knowledge about electrochemical mass transport at the beginning of its
application (1950). The same authors developed correlations for predicting
the limiting current in laminar free convection at vertical electrodes, as well as in turbulent flow between cylindrical electrodes, with the inner cylinder rotating and the outer one stationary.
Ibl 8 reviewed early work on free convection, to which he and his co- workers contributed notably by the development of optical methods for the study of the diffusion layer. 101 ' 7 ' 9 A discussion of their application falls outside the scope of this review and has been given by Muller. 104 102 In a survey of 1963, Ibl lists 13 mass transfer correlations established by the limiting current method. Only 4 of these were known from quantitative theoretical predictions. The total number of publica- tions since 1963 has almost quadrupled (references 1-88) The majority of these concern flow situations where theoretical predictions are at most of a qualitative nature An increasing number of publications deal with model studies of complex situations, e g , packed and fluidized beds, and with hydrodynamic studies. 14
Part II. Review of Applications
For the purpose of this review we will distinguish experimental
studies of mass-transfer in: (a) simple laminar flow; (b) laminar flow with
complications; (c) turbulent flow and mixed convection; (d) stirred vessels; (e) cells with gas-evolution; (f) particulate flow systems.
A related technique of mass transfer measurement will also be discussed briefly.
1.4. Simple laminar flows
For the flow situations listed as number. 1 through 5 in Table 1.3 the mass transfer rate can be derived directly from a solution of the equa- tion of convective diffusion. The velocity profile near the electrode is known, and the equation is simplified by appropriate similarity transforma- tions. Newman 103 has reviewed these solutions in detail. Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and experiment
In the case of the rotating disk there is a uniform axial velocity towards the disk, which depends only on the normal distance from the disk, so that the mass transfer rate is also uniform. If the center of the disk is non-reactive, the flux to the active ring is higher than itotherwise would be, and the electrode is no longer uniformly accessible. The maximum rate is found at the inner edge of the ring, where radial convection is no longer uniform. Thus, the maximum is due to the non-coincidence of the uniform hydrodynamic boundary layer and the diffusion layer (relaxation process). 15
Table 1 3 Mass transfer correlations obtained by limiting current measure- .ments.
Sy tee C rrelation Range Paraaeters Electrolyte Authors
Ia Rotating disk Mo0.62Re1125e3 Ree2.7n105 rradiu, 0 2 Siver.kabanov 87 Recr2/v r. =inner radius - Cu" Hsueh.Newman 98 Nu-k/D router radius Fe(CN) Okada, et Neoson,Riddiford 88 •Daguenet 59 b Rotating rang 3uNud xs k r ( 1 _r/r 3 ) 2t3 Rea2.7x105 Daguenet59
2 Coaxial flow (aNu1.62(ReSc d/L) 1 l.6l0< 2 d r -r. 0 2 tin, et Al in annular r /ro I O. S Rest d/L (b)NU1.94(tteSc d/L)1'3 Cu Ross,Wragg 36 r/r0.S ReSc d/L<9x10 7 Wragg,Ross 63 3 Parallel floc Nu'1.85(ReSc AlL) 113 75 4 7((,5)l4 Free cx nec (a)Mo 0 S 306 (GrSc S 1012 Lh ght Cu 2 wagn 1 tion at a Grglapt 3/v2p SO0 (bf °Nu..0.45(GrSi) 1 " 4 9xl09cGrScn4.bolO V=electrode velocity Fe(CN) Fouad,Gouda 31 2350'cSc<3650 Ltraxsfer length .5 Moving Nu-1.13Pei/ 2 40005 /Pe'C40000 - Fe(CN) Rotte7 3 Continuous PevL/0 Sc>700 ROtte,iaoxiarx74 cylinder Nuwkk/0 See however page 17 of text. txpectrd constant is 1.939 for ri/n 0.5 (cf. eq. 1-6 text) See page 23 of text. Expected constant is 067. (continued on p 26) 16 A maximum in the mass transfer rate at the entering edge of the electrode occurs in all cases where the main flow is parallel tothe electrode. This is the case in flow through pipes and channels, but also, e.g. , at a flat plate, where the velocity near the plate has a transverse component. In all such cases the parallel velocity component close to the electrode can be considered linearly dependent on the distance from the electrode. This is justified by the extreme thinness of the diffusion layer compared to the region of velocity variation (large Schmidt number). Similarity transformations lead then to two types of correlations: the boundary layer type, 1/3 = C.Re 2Sc , (1.4) valid for boundary layers which have transverse velocity components; and the Lvque type, • Nu=C.(ReSc d/L) 3 • • (l.5) valid for parallel flows. The rotating disk correlation Ia in Table 1.3 • is an example of the first. Numbers 2 and 3 belong to the Lvque type. All are characterized by the exponent- 1/3 on the Schmidt number, correspond- ing to a ratio of boundary layer thicknesses (hydrodynamic/mass transfer) of the same order of magnitude: Sc 3 The correlation of Table 1.3, for mass-transfer at infinitely extended parallel plates, to which Hickman's 32 experimental channel is 17 a good approximation (aspect-ratio 10 to 15), is a special case of that in a 103 annular channels. The general expression for such channels is: Nu - 1 615(ReSc d/L)113 , (1.6) where is a function of the geometric paratheter.K the ratiobetween inner and outer radius. Re and Nuare based on the equivalent diameter, i.e.., the difference between inner and outer diameter. Figure 1.1 shows 1/3 as a function of K. The limiting case K - 1 (r =r., d = 2h where h is the channel 0 1 e height) gives . . Nu = 1 8488 (ReSc d/L)'3 , (1.7) which is identical with the expression first found empirically by Norris and Strei 32 for heat transfer in flat rectangular channels, and .confiried by Hickman, with a scatter of about 7% around the theoretical values. 2 The correlation 2 of Table 1 3, established by Lin a o for coaxial flow in annuli with K = ' 0.5, was originally compared by them with the other limiting case of (6), K - 0 (r. ± 0), i.e., mass transfer to the wall of a circular tube: Nu = 1 615 (ReScd/L) 1 " 3 (1.8) This procedure was criticized by Friend. and Metzner 128 who calculated that the constant should actually be 1.939 for the radial ratio 0 5 Part of the 19% discrepancy between the experimental results and the theoretical prediction can be ascribed to the use of incorrect diffusivities For the experiments using the ferri-/ferrocyanide redox reaction an estimate of the • .•• •. • • • ••• •• • •• •• .•. • • •• • • •. • •• • .• • • • • . • •. • • •.. • • • •..• •.. • • • • -. •- • . • • • 18 • •.: • • •. • • • • ••••• • :.. •• •.. . • • • •• • • • • • ••. .•• . • ••. • •. • : H 19 error is possible. The value of the mobility product iD/T, based by Lin c.s. on the mobility at infinite dilution, is 6-20% lower according to more recent measurements using the rotating disk (The concept of the rotating disk integral diffusivity will be discussed below). Table 1.4 lists these values. Also given are LID/I for oxygen in NaOH as used by Lin c.s., and according to recent measurements by Davis a.o. 52 in KOH. The latter value is no less than 25% lower than that used by Lin c.s. Since, according to (6) the mass transfer coefficient k = constant x D 2 " 3 a discrepancy of approximately 31% in the diffusivities would be necessary to cause a deviation of 19% in the constant, assuming all other properties unchanged. Consequently, the discrepancy between the results of Lin c,s. and equation (3) cannot be attributed entirely to the use of incorrect diffusivities. The theoretical relation (6) was tested also by Ross and Wragg 36 63 in annuli of the same radial ratio 0.5 as used by Lin c.s. , and a few other ratios. They used the copper deposition reaction in acidifiedsolu-. tion, the solutions being 0.01 or 0.05 M in CuSO 4 . The expression original- ly presented to correlatetheir data fell 10% below the theoretical one based on (6) In a later review the data were re-correlated using rotating disk integral diffusivities measured by Arvia a o , instead of the older diffusivity data of Cole and Gordon from diaphragm cell measurements The 4 latter had been expressed by Fenech in the form of a LID/T correlation with weak dependence on CuSO4 andH2SO4 concentration. The valueof LID/I found by Arvia a o for solutions dilute in CuSO 4 (and not containing glycerol) reactant K3Fe (CN) 6 equimolar K4 Fe (CN) 5 equimolar 1method + excess KOH NaOH KC1 KNO3 KOH NaOH KCI KNO3 Lin 2 2.67 2.67 2.67 2.67 2.31 2.31 2.31 2.31 LM Eisenberg97 2.50 1 2.15 cc Arviá51 2.78 252 2.00 2.52 2.09 1.75 RDE Gordon47 2.34 + 0.0141 1.87 .+ 0.0341 RDE * Van Shaw 22 2.30 LFA * ROtte 53 2.27 DS 173 * Sih 1.49 RDE Smyr1174 1.81 RDE reactant 02 in 0015MKOH 2MKOH Lin 2 7.37 1CT Davis 2 5.40 4.36 RDE,CC Gubbins 120 5.46 5.60 POL based on measurements with one composition only CC capillary cell DS diffusion to a sphere ICT International Critical Tables LFA Laminar flow annular cell (Lvêque relation) LiVI Limiting mobility POL Polarography RDE Rotating disk electrode 21 was 20% lower. Re-correlation on this basis yielded results very close to the theoretical expression. 63 These two tests of mass-transfer theory by means of the limiting current method point to the importance of correct diffusivity data to be used in correlating the results. From an analysis of the mass-transfer process, which will be discussed in Chapter 2, it can be shown that • . diaphragm cell measurements do not necessarily provide the best effective diffusivity values to beused in limiting current calculations. A theoreti- caly correct diffusivity to be used in multicomponent solutions cannot • be defined or constructed from binary. diffusivities. The only valid statement that can be made is that, in the limit of infinite dilution, this diffusivity must be identical with the Nernst-Einstein diffusivity calculated from the limiting ionic mobility: RTA. D.R Tu i = 1 0 • 2 1 zF Early experimenters were therefore forced to use either diaphragm cell data, scarce for multicomponent systems, or limiting diffusivities, which were known to be too high. In the face of this dilemma later investigators increasingly resorted to .the use of effective, integral, diffusivities. Capillary cell diffusivities are .attractive because of the convenient method of measurement but from a viewpoint of diffusional similarity the rotating disk integral diffusivity is more appropriate for work in convective mass transfer. .. The use of rotating. disk diffusivitjes means, in effect, that a mass transfer correlation for a well defined coñvèction profile, once its 22 validity established by theory or independent experiment, can be inverted to calculate effective diffusivities from (and for) limiting current measurements in the same regime and the same electrolytic solution. This stratagem could, in principle, be applied quite general- ly but it has, so far, been restricted to rotating disk mass transfer. In Chapter 2 the concept and applicability of these diffusivities will be discussed, and in Chapter 3 such diffusivities are calculated for the Cu2 ion in acidified cupric. sulfate solutions. Diaphragm cell data have been used all along because they were the only ones available, e.g., the effectiveness of the limiting current method for mass transfer measurement was demonstrated quite early by the results of Tobias and co-workers in free convection.at vertical electrodes (number 4 in Table 1.3). Excellent agreement was obtained with the theoretically predicted correlation. These experiments were carried out with the copper. deposition reaction, which provides a large density difference at high concentrations. The diffusity data used were diaphragm cell data of Cole and Gordon. These data, correlated by Fenech, 1 ' were also used in later work on horizontal electrodes. The important question whether the Cole and Gordon data were adequate.for the work in question, provided a key motivation for the study of rotating disk integral dif- fusivities in the CuSO4 -H2SO4 system, to be reported in chapter 3. *In this connection it should be noted that the most careful studies 88 ' 59 undertaken to confirm the rotating disk correlation (la and b, Table 1.3), used the iodine reduction reaction with very small reactant càricentrations, and employed diaphragm cell diffusivities of K1 3 in KI. 23 Later measurements by Fouad and Gouda, 3 ' who used the ferri-/ ferrocyanide redox reaction, yielded a much lower coefficient in the free convection correlation 4 of Table 1.3. However, as theoretical calcula- 113 tions of the migration effect in free convection show, the small density difference driving the convection in the case of the redox reaction, in combination with the presence of highly mobile 0H ions, tends to destabilize the velocity profile. It is not unlikely that this effect, and the uncertainty about the interfacialcomposition, are responsible for the discrepancy between the result obtained with the copper deposition system andthat with the redox system. The application to free convective mass transfer is a particularly severe test of the limiting current method because the driving density difference, which enters intothe correlation: Nu = C(GrSc) 4 by way of the Grashof number: Gr=tpgL 3 / 2 p is influenced by the accumulation or depletion of supporting electrolyte at the electrode. Thus there is an interaction not only of diffusion and Lonvection, expressed in the coupling of the equation of motion and the equation of convective diffusion, but also of electromigration and convective diffusion This is true for well supported electrolytes, and a fortiori for solutions concentrated in CuSO4 as used frequently in free convection studies The situation has been analyzed in detail elsewhere 113 • 24 In' free convective mass transfer at electrodes, as well as in forced • convection, the diffusion layer extends only.'over a very small part of the hydrodynamic boundary layer. Whereas in laminar forced convection the ratio -1/3 d'h was on the order of Sc , it is in laminar free con- vection: Sc 4 - dh There is an important qualitative difference, too, in that there • is an inner part of the boundary layer, where the velocity indreases to a maximum determined by a balance of viscous and buoyancy forces. The diffusion layer. thickness, in fluids of high Sc, is of the same order of magnitude as this inner part of the boundary layer. In the outer part of the boundary layer, where the velocity decays, the buoyancy force is unimportant. - Again different as far as concerns the velocity profile in the region of varying concentration, is mass transfer at a moving continuous electrode, exemplified by correlation .5 of Table 1.3. Next to the uniformly moving • continuous flat plate, the flow generated by a moving, continuous (e.g;, extruded) cylinder is the simplest of a class of laminar boundary layer flows first investigated by Sakiadis. 129 In high Sc number liquids the • diffusion layer is contained in a region near the moving electrode, where the liquid velocity is practically equal to that of the cylinder. The mass transfer rate should follow then a penetration type correlation, • involving the Pclet number (vL/D, where v is the electrode velocity;. L the transfer length): 25 Nu /)Pe 2 (1.10) Experiments where the electrode is circulated into and out of the solution are difficult to realize without electrolyte and current leaks. Rotte, 74 using a nickel wire circulated through ferri-/ferrocyanide solution, found only approximate agreement with equation 10; mass transfer rates were mostly too low and showed considerable scatter. 1.6. Laminar flOw with complications It is remarkable that the well-known boundary layer flow along a flat plate has not been used in electrochemical studies The expected equation NuL = 0 677 ReU2Sc"3 (ReL Nu = Nu 1. )1/3 x x,plate (1 - (1.12) Their results, however, were still 24% too high compared with this correla- tion. The velocity profile must have been steeper than would correspond to flat plate theory. This was ascribed to the merging of the boundary layers * Numbers 6 through 20 of Table 1.3 26 Table 1.3. (continued from p. 15) System Correlation Range Parameters Electrolyte Authors 6 Rotating and no correlation -30vv-e3000 - h-distance disks Cu Qiada, et al. stationary r-óc. r.rsdius i stat ~ l rot o disk in h.0.19-.2cm restricted housing • 1 Developanp no general 825.vRe<3000 1-entrance Fe(CN) Wranglen,Nilnson 16 flow in correlation d/10.343 length • rectangular k-.v 1 " 2 dIh-1.14 d-eq.diameter - channel 1-transfer - (L e/d 84 Developing P6J_ 3 . 647 RCV 2Sc ''3 Ltrannfer Fe(CN) 6 Bazan flow in length • annular channel (L/d<0.OIRe). - 3/4 b) deq.diter Satan 35 9 Developed Nu-1.43(ReSc d/L) 3 d/L.8.1 or 9.2 d-nq.dianeter Cu Pa1e de Re-ed/v • flow over 1.0.15cm 1-transfer Iribarne et al. 78 isolated 4xlO5sReSc d/Lol.6s10 7 length • strips 10 Falling No-1.21(ReSc d/L) 1'3 Rer/lo.g6 3 /3v 6-file thickness Fe(C..fl ribarne,Gosmanet al. 56 lIquid PPu-kd 0 r-nass flow rate film 70'Rev700 1400 Transoerse 13 • lla (a)Nu-l0RRe 044 5c 3 lb0vae<1000 d-diameter Fe(CN) Gransoan,Iblet al flow around d/d toienO• 2 cylinder cylinders (b)Nul.20ReO.42Scl/ 3 d/db vO.l .Rc-ed/v • See, hovever, 27 of text. p. Eq. 1-12 tent gives the eapected correlitcod. - Results of lib appear to be more generally applicable. (continued on p. 31) 27 Bazan and Arvia34 ' 35 also used equation (11) to correlate results obtained in internal flows, this time in annuli with insufficient hydro-, dynamic entrance provisions. Some of their data fitted the form of (11) with only a slightly smaller coefficient. There appears to be no reason why equatiOn (11) could not be tested ma flow channel of sufficient height provided with a suitable flow- straightening section. The results should be relevant to many corrosion processes occurring in maritime structures. The Lvque mass transfer model has also been tested for flow of a liquid film under gravity (10 in Table 1.3), where it should hold for Reynolds numbers in the laminar range in spite of the presence of surface waves on the film. The latter caused small local oscillations of the mass transfer rates, on the order of 2% in the laminar range but higher in the turbulent range which was also investigated (see below) In these measure- ments Iribarne c.s. 56 found rates slightly below those predicted by the Lvque relation, but the difference is not significant. Of considerable, interest is the use of small isolated electrodes, in the form of strips or disks embedded in the wall, to measure local mass transfer rates, or rate fluctuations. In fully developed channel or pipe flow, the mass transfer to such electrodes according to the Lvque model is given by k avP 0.8075(D 2S/L) 1 (1.13) • where S is the velocity gradient at the electrode and L its length. Equa- tion 6, 7 and 8 are specific forms of:this.general relation.'. 78 Iribarne a o measured k at an array of electrodes each 1.5 mm 28 long in the flow direction and 1 cm wide, spaced 5 mm apart along the wall of a rectangular channel. The actual rates found were considéràbly below those predicted by equation 7, even if allowance was madefor the rectangu-. lar shape of the channel cross-section. Considerable scatter characterized the data. Probably the electrode spacing was not large enough to assure decay of the concentration profile in between electrodes. Interestingly, the use of such extremely short sections may complicate the Lev'que mass transfer model which neglects longitudinal diffusion. At the leading edge of electrodes this assumption is not correct, and the flux is actually increased, over a distanceof the Order of d e/viSc. Newman 13° treated this problem for the case of pipe flow and obtained for the local mass transfer rate, in the limit of very small transfer lengths: : NU/NULvque = 0.5519 Pel2(h/'6 (1.14) For a Pe number of 10 a typical lower limit in laminar flows, and a channel diameter of 2 cm, the region of axial diffusion is 0.2 mm and the average rate in this region is roughly 10% larger than according to(l3). The influence of axial diffusion will not normally be felt when larger Pe numbers are involved The use of strip or spot electrodes has been üveloped extensively by Hanratty and co-workers, who used the in particular to record mass-transfer fluctuations in turbulent flows They used disks of diameters 0.38 to 1.63 mm (effectively equivalent to rectangles of L equal 0.82 times the disk diameter) and rectangular shapes 0 51-1 57 mm wide, and 0.076, mm to 0.53 mm long in the flow direction Since they were interested principally in the intensity of the mass transfer fluctuations, the coefficient in (6) is of less importance. From the mass transfer intensity one can obtain information about the turbulent velocity intensity near the wall. In turbulent flow the diffusion layer is an order of magnitude smaller than the viscous sub- layer, at least for electrolytic solutiosn. Therefore, the velocity fluctuations as well as the average velocity transverse to the wall are negligible compared to the longitudinal quantities, and the Lvque model applies again, but with a time-dependence. For small frequencies, f, Hanratty and Reiss 19 ' 2° deduced a simple relationship between fluctuating mass tansfer and velocity gradient, in terms of the dimensionless frequency: - 2 fL 2 ''3 • by means of which the velocity intensity can be calculated from the mass transfer intensity and the spectral distribution function of mass transfer fluctuations. By thus measuring and correlating mass transfer fluctuations at strip electrodes in longitudinal and circumferential arrays, Hanratty and co-workers 19 ' 20 ' 22 ' 27 ' 49 obtained information about the structure of turbulent flow very close to the wall, where hot-wire anemometer techniques become unreliable A concise review of this work has been given by Hanratty. 50 When electrically insulated strip or spot electrodes are embedded in a larger electrode, and the turbulent flow is fully developed, the steady mass transfer rate gives information about the eddy diffusivity in the viscous sub-layer very close to the electrode (see section 6, below). 30 The fluctuating rate does not give information about velocity variations and is markedly affected by the size of the electrode. The longitudinal, circumferential and time scale of the mass transfer fluctuations lead Hanratty50 to postulate a surface ienewal model with fixed time intervals based on the median-energy frequency. The basic Lvque relation (13) was also used by Hanratty and co- 89,67 workers to study surface velocity gradients in laminar and separated flow around a cylinder, in the ranges 60 < Re <. 360 (where vortex shedding first occurs) and 5000 < Re < 100,000 (region of constant drag). Local velocity gradients were measured by embedding strip electrodes in an inert cylinder surface and measuring the timeaveraged •current. Local mass- transfer rates were also measured, with the entire cylinder surface as cathode and insulated disks as measuring cathodes. The recorded rate- distribution over the circumference was in good agreement with the shear- stress distributjon calculated from the measurements on the Inert cylinder Only at very low Reynolds numbers did free cOnvection interfere with the velocity field, in particular at the forward and rear stagnation points. Also, velocity gradients at the rearand near the separation point are small, so that (11) becomes inaccurate. The separation point was determined with an accuracy of within 10 by the use of twin strip electrodes of length 125p, separated by SO-7011 insula- tion. The electrode giving a diminished signal when the other one is activated, is in the wake of the other. With a proper choice of electrode configuration, this procedure can be employed to measure quantitatively the direction of surface velocity gradients in three-dimensional boundary layers. 64 The extreme care with which embedded eiectrodes for such 31 Table 1.3. (continued from p... 26) System Correlattor, Range Paramete 5 Electrolvte Authors llb Nu 0 38Sc 0 2 S Re 100 d-d aneter F CN) ogtl nder B ii 20 (c0.56Re05,000lRe)5c0 ' u000sc 12 Flow through a (a)Nu-064Sc '5 IORe050 - d.diaaeter Fe(CN) Vogt1Onder,Bakker gauze ..0.S2Re 112 Sc 3 I3000Scu2000 gauze wire open area 66% e.free velocity (b)Nu.l.o3sc hi'S,oblge h/ sc h/3 open area $66 Rewvd/v 13a Free conveCtion Nu0.53(GrSc)1 1.2ol07 13b No 0 23(GrSc) 0 2 3 i06 Gr ScI10 Fe(Cl) Beder 54 14 Free convection Nuw240.59(GrSc) 114 2.3xl08uGr5c 15 Free Convection 2xlO5cGrSc<501010 Nu0.45(GrSc) 11 ddiameter Fe(C%) Biagg 61 at a vertical Nu,Gr see 13 . gauze wire gauze - - -. 16 Free c nvect 0 N see Fig 4 0 6 0 1ength(hczgtt) F (CN) Ru oc et at a vertical 'rt. L/R(GrSc) "4 2xl0tGr5ci6xj0 12 Rradius . 17 Combined Nu=0.636(G r Sc )L' 4 4xlO7 tion at a operating body forces . . vertical plate (continued on p. 38) 32 • hydrodynamic measurements have to be prepared, is demonstrated in a paper IME by Son and Hanratty. 67 Altogether, these. investigations are a remarkable example of the versatility of the electrochemical mass transfer model. Implicitly the method is considered reliable enough that generalized correlations, e.g., (13), can be used in the reverse manner, to measure convection profiles at a reacting surface. . -In' their work on cylinders, Hanratty c.s. did not measure over-all mass transfer rates, which are important from a chemical engineering view- point. For cylinders in transverse flow this had been done before, by 13 . .... • 5 Grassrnann, Ibi and Trueb, and by.Vogtlander and Bakker. (The former also were the first to report measurements of local rates, by means of a strip of 0.45 nun length embedded in the, cylindrical cathode: They reported a minimum at the forward stagnation point, not confirmed by Son and Hanratty measurements and perhaps caused by free convection). The results of Grassmann c.s. are somewhat higher than those of Vogtländer and Bakker, who measured overa lower.range of Re numbers and fitted their results tóan empirical heat transfer correlation This apparently accounted well for free convection contributions, by the presence of a term in Sc only. Also, the wires used by Vogt1iider and Bakker were very thin compared to the flow tube, diameter, while this was not the case in the others' measure- ments. The expressions 12 (Table 1.3) forgauzes are 10 to 20% higher than those.for a single wire, if based .on the approach velocity. Free convection flow around horizontal cylinders is also laminar in nature, for moderate values of GrSc, and follows a relation of the same 33 type as that for vertical plate electrodes Nu (GrSc)4 (1.16) Schitz 24 observed local transition above GrSc = 3x10 8 , by the use of an insulated strip electrode. The correlation obtained by Weder 54 has a somewhat higher exponent, interpreted as an indication of local transition at the highest GrSc numbers dealt with Wragg 61 investigated free convectiofl mass transfer to vertical gauzes and found likewise good agreement with an expression similar to (16). For free convection, at a sphere Schtitz obtained the correlation: Nu = 2 + 0 59(GrSc)'4 , (1.17) which takes the effect of pure diffusion into account by means of the constant. The flow here is not laminar, according to his measurements with local spot electrodes, but already in transition to turbulence for the GrSc range inquestion. Free convection at vertical cylinders differs from that along vertical plates because of the influence of curvature The effect is important only if the diffusion layer thickness is no longer small compared to the diameter Since the diffusion layer thickness increases with x, this can be expressed as • x/R - O(GrSc) 1/4 where x is the vertical coordinate and R the cylinder radius Ravoo a o 34 .1 • • .• • •• •• • •• . • • .• . • ¶ . - . . - • .••• . developed an approximate solution for this case and, using wires of diameter, 0.1 to 2.64 mm, length I to 374 cm, found fairly good agree- ment with the theory. (Fig 1.2). Another case of free convection with some complications, but amenable to solution, is that due to combined temperature and concentra- tion differences. Marchiano and ArvIa 66 tested a theoretical solution for mass transfer in such a laminar flow along a vertical plate, which employs .a combined Grashof number as follows: Gr = {(ac) 314 + (T)3/4()h/4}4/3 gL , (1.18) where cv is the concentration densification coefficient and C that due to temperature. They found fair agreement except for an unstable flow region if the temperatur.e densification was of opposite sign but not large enough to cancel the concentration effect. For stronger cooling of the cathode the flow became stable again and mass transfer rates satisfied the theory. Earlier, de Leeuw den Bouter a.o. 65 had already experimented with such combined free convective transfer, assuming complete analogy iftheGrashof number employed is: Gr ={cthc AT()h/2} &L (1 19a) Their results for cooperating body forces come very close to a confirmation of this complete analogy, since the expressions (18) of Table 1.3 are also valid for the heat transfer rate which was measured simultaneously 36 * 0252 Nuh = 0.64(GrhPr) . (1.20) where Grh = {AT + cLc(i-) 2 } &!-2 (1 19b) The expression (18) appears to have a weaker theoretical basis, too, since it approximates flow profiles by a single quadratic function of the distance. A more complicated case of combined thermal and dif.fusional free. convection is that around a horizontal cylinder. This has been investi- gated by Weder, 54 who likewise used a combined Grashof number of type (19) but corrected for the higher exponent found in isothermal mass transfer (13b of Table 1.3). The correlation finally obtained is: . . *0.28 . .. Nu = 0.23(Ra ) . . . . .. . (1.21) 112 with Ra* = (GrSc)1 07 + 1. 49 (GrhSc)( .E ) (1.22) Weder's experiments were all carried out with opposing body forces, and large current oscillations were found as long as the (negative) thermal densjficatjon was smaller than the diffusional densification. (Note that the Grashof numbers in (22) are based on absolutemagnitudes). Local mass transfer rates oscillated by 50%, and total currents. by 4%. When the thermal densification dominates, the stagnation point moves to the other side of the cylinder and the boundary layer, which separated in purely diffuional free convection, remains attached. Free convection in restricted spaces has been investigated by Bhm, 90 Ibi and Frei, 48 and earlier by Schmidt The latter author restricted the .37 distance between parallel plate electrodes in halogene redox reactions (without supporting electrolyte) to such small values that convection could hardly develop Similar experiments, also using closed cells but with larger electrode distance were reported by O'Brien and Mukherjee 30 B5hm c.s. did restrict the liquid movement by varying the distance between vertical cathode and vertical diaphragm from 6 to 0.15 mm, but did not enclose the electrolyte between these. Various experiments with impeded inflow andoutflow were also reported. A critical distance between vertical electrode and diaphragm exists, below which the current is less than for a single vertical plate with distant counter-electrode This critical dstance is given by: . = 5.48L(GrSc) -0 . 3. (1.24) Conspicuous is the high exponent on the GrSc number in the mass transfer correlation (20) of Table 1.3. Since the current is almost independent of D, this must mean that the reacting ion is depleted at the downstream end of the narrow slit between cathode and diaphragm so that the total current is largely determined by the convective transport of reactant .into the slit, which depends on the density difference but not on diffusivity, Investigations like the one just discussed a.revery important from an engineering point of view.. Mass transfer processes rarely take place in the simple configurations leading to the classical correlations for forced and free convection. By examining the influence of flow obstructions and . secondary flows, one can gain insight to what extent practical mass transfer situations deviate from the theory On the other hand, the 38 Table 1.3. (continued from p. 31) Syste. Correlation Rang. Pa.ri..tens Electrolyte Authors 0.243 18 Sc) - 6zlO900rScc2 lO L.height Cu 2 de Leeuo den Router, opposing and for 2ol06 - Cr see eq.(1.19) text 19 Conbined Nu.0.23(Ra) 028 2.340600rScol.5xl010 d.djametey Fe(CN) Weder 54 thermal and Rasee eq(21)tent 5.5alO300rh Pr<9,7x7010 b cylinder diffusional opposing body forces 2xl0 20 Free convection NuL. O. 022 S(a/L) 2 (GrLSc) °85 25l06 cGrSc.cZxlQ8 a=distance Cu 2 Bb1, et al. 48 at vertical 0.005 21 Mass transfer Nu'0.276Re119(sc L)1/3 50 22 Mass transfer Nu1.080(ReSe 6fL) 113 Rei'103 Sfil. thickness Fe(CN) Iribarne, cc al. 56 entrance Re'g*5 3/34 L/65900 w'g6 3 regime in - turbulent • - • falling film (continued on p. 39) 39 Table 1.3. (continued from p. 38) System Correition Range Parameters Electrolyte Authors 23 Mass transfer Su0.023Re 08 5c 21000 Re'30,000 d.Z(r , -r 0 Un, et al. 2 fully 300 24a 2 . k0.11SSc 3 " 4 5000sRet50,000 Cu Sci.ijtz 38 (pipe) (Deissler) Sc2170 24b Nu.(f/2)ReSc113 70000Re<60,000 d.eq.diameter FelCN) Fhbbard,l.ightfoot 42 - (channel) (Chilton-Colburn) 17000Sc<30,000 26 k.O1159c 314 Reol000 4fil. thick Fe(CN) lribarne 56 (falling (Spalding-Van Dr,ent) 1000 27a Fully developed NuO.00776Re 04 Sc 3 2.7xlO5 vRenlSalO rotati0n rate 1 3 Daguenet 59 turbulent floa Re-Wr 2 /V 34555c<6450 rradius disk at a rotating disk b N.. 0 0llRe0 8916 Sc 0.249 8 9 10 5 28 Fully developed Nu.Nud.k 2.7xlO5ngeclSnlO6 rinner radian l Daguenet 59 turbulent flo.. 345 (continued on p. 46) 40 blind labor involved in copying and investigating one or another industrial configuration is avoided. Rather surprisingly, research on forced flow iiirestricted geometries is also scarce. An example is work by Okada a.o.' ° on a rotating disk flow where the electrolyte is completely enclosed between the rotating disk and the counter-electrode. Mass transfer was measured at the rotating as well as at the stationary disk The distance between the disks was varied. At low rotation rates the flux at the rotating disk was higher than predicted by. the Levich equation (1 of Table 1.3). The flux at.the stationary disk followed a relation of the.Levich type but with a constant roughly 2/3 that of the rotating disk equation. . Flow patterns in agitated vessels can be profitably studied start- ingwith disks and cylinders rotating freely in restricted cells. Investi- gations of this kind will be reviewed below, after a survey of the better defined turbulent flows. * 1.7. Turbulent flows and mixed convection . Forced convective turbulent mass transfer relations are of interest for two reasons: their practical impOrtance since turbulence efficiently increases transfer rates; and the insight one can gain from them into the mechanism of turbulence very close to the wall, where the usual hydro- dynamic probes (hot wire anemometer) disturb the phenomenon to be theasured. A variety of turbulent mass-transfer relationships are listed in Table 1.3. The most important ones are those for channels and tubes, but several other ones have been investigated for, their hydrodynamic interest. Turbulent flow theory is not abre to predict mass transfer rates quantitatively, except in the region of developing mass transfer where a Numbers 21 through 37 in Table 1.3 . . . . . . 41 Lvque type correlation between rate and friction coefficient can be established: 0 8o75()hI3 or (1.25) k/v = k = 0 8075 L1'13Sc2" , (1.26) where subscript + denotes quantities made nondimensional with respect to the wall shear stress T by the 'fiction velocity" v - (Tip)112 = 1/2 By substituting the well-known Blasius relation for the friction factors, equation 21 of Table 1.3 is obtained, which van Shaw, Hanratty and Reiss 22 tested by limiting current measurements on short pipe sections. They found Re and (L/d) dependence according to the theory but rates on the average 7% lower than expected In a later publication 50 this was traced to incorrect flow rate calibration Schutz 38 also found good agreement with this mass transfer entrance equation for small values of L/d. The condition for validity can be expressed as L < 10. Iribarne, Gosman and Spalding 56 showed that the Lvque relation is also valid for turbulent flows in films, with the same entrance condition and Re > 10 When the concentration profile is fully developed the mass transfer rate b:ecomes independent of the transfer length. Spaldiñg 131 has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular + eddy) viscosity and total diffusivity possess a universal character. In that case the transfer 42 rate k+(see 26) can be expressedas a single, universal function of the transfer length L and fluid properties, expressed in a molecular and a turbulent Schmidt number: • = k(L, Sc, Sc) Equation (26) is the asymptotic form of (27) for small L. The specific form of (27) for larger L can only be derived with the help of • a particular hypothesis about the distribution of.the eddy diffusivity (or viscosity) in the turbulent boundary-layer, in particular near the wall, • in the inner part of the viscous sub-layer. Since the mass-transfer rate attains a constant value for very large L (>10), the dependence of this asymptotic k , on the Schmidt number has been the objective of numerous • experimental investigations, whereby 'the limiting current method had decisive advantages over earlier dissolution methods. An equally important objective of these turbulent mass transfer • measurements was the establishment of the •exact dependence of k on the Reynolds number. According to the well-knowTl empirical Chilton-Colburn equation for turbulent transfer one should have: = . ReSc1'3 Nu (1.28) • which corresponds to an additional dependence of k ~ on the Reynolds number. Semi-empirical modifications of the Chilton-Colburn equation have been proposed, 132 which also imply this additional dependence. Table 1.5 lists some expressions valid in the limit of large L and high Sc. • Son 90 has reviewed the experimental evidence from electrochemical and other model experiments. He concludes that the eddy diffusivity varies with 43 Table 1 5 Expressions for fully developed turbulent mass transfer. Author Expression Basis Chilton-Colburn Nu = X/2ReSc 3 empirical k. = /X/2 Sc 23 Deissler' 33 Nu =0 0789/A ReSc4 Dt - k = 0.112 Sc 314 Lin, Moulton, Nu =0 057/A ReSc 3 and Putnam4 k = 0.081 Sc 2' 3 Vieth, Porter, Nu = 4.586(u max /u avg2)eSc 3 semi- and Sherwood' 32 k4.586 u /u 1'4s2'3 empirical + max avg z Spalding 131 Nu = 0.053 A ReSc semi- k = 0.0746 Sc 3"4 empirical 44 the fourth power of the distance from the wall, assuming that the friction factor takes care of the Reynolds number dependence Hubbard and Light- .42 . . 1/3 foot earlier concluded to a Sc dependence on the basis of measurements in which the Schmidt number was varied over a very large • range. These did not exclude a lower Reynolds exponent than 0.88, and re-affirmed the value of the classical Chiltón-Colburn equation for practical purposes By more refined measurements using strip electrodes at different orientations with respect to the mean flow, Sirkar and Hanratty 69 later showed that the transversevelocity fluctuations playa significant part in the turbulent transport very close to the wall, and that the eddy diffusivity may indeed be dependent on the cube of the distance y, lead- ing to a Sc dependence of mass transfer correlations. The laminar boundary layer on a rotating disk becomes turbulent at Reynolds numbers above 10 Mass transfer to disks and rings in turbulent, or almost completely turbulent, flows has been measured by Daguenet 59 . and by 81 . 1 / 4 Ellison and Cornet. The lat.ter found a dependence on Sc , in accordance with Deissler's model (Table 1.5) Daguenet, 59 who used a higher Sc range, concluded to agreement with the Lin-Moulton-Putnam model, with a dependence on Sc "3 Summarizing, it can be said that a conclusive investigation of the turbulent transport process is still extant, and that it will be necessary to synthesize inferences from optical as well as mass transfer measurements • • The latter will have to be made at very high Reynolds •nunibers (>50,000 in • channel flow) and at very high Schmidt numbers (>10,000) to yield critical information about the transfer process. • 45 Flow between rotating cylinders has been referred to in the historical introduction. (1.3) since it was recognized early that Couette flow can be realized, if the inner cylinder rotates more slowly than the outer cylinder, or is stationary.. Experimentally this is hard to realize, and so far only work with a rotating inner cylinder has been reported, where Taylor instability occurs at low rotation rates. The other inherent advantage of concentric cylinders is the uniform current distribution, which makes this model extremely attractive for applications.. The over-all results for the turbulent regime are in good agree- ment with those obtained by dissolving cast metal or salt cylinders.' 18 ' 119 Rotating cylinders have also been used to measure the influence of surface roughness, on the turbulent mass transfer rate.' 21 . Turbulent free convection has early attracted attention in limiting current studies, because the laminar free convection boundary layer becomes turbulent at a certain height, approximately characterized by the Rayleigh number . . RaL = GrLSc Z loll to 10 13 . (1.29) Fouad and 1bl 12 have suggested that the transfer rate in free convective turbulent flow is expected to follow a Ra dependence. Fouad and Ibl 12 found a dependence 'slightly lower, probably due to the partly laminar 'regime on such plates. Because the exponent is not fixed bytheoretical considerations, there is considerable variation in the coefficients of turbulent free convection correlations (Inaccuracies in the calculation of the density driving force may also be responsible for this, in particular in the ferricyanide reduction reaction) 46 Table 1 3 (continued from p 39) System Correlation Range Parameters Electrolyte Authors (a)Nu1..0.079lRe07Sc0356 .29 Turbulent 103 (b)Nu.0.079lReO7ScO0hS6 10 3 - (did)° . Sc.2450 . . diameter 0.221d./d <0. 68 . (stator) 30 Transotional . . and turbu- . lent Couette flow between rotating cylinders 31a Turbulent Nu0.31(GrSc) 028 4xl0 13(CrScCl0 L=height plate Cu2 ' Fcuadlbl 12 free convec- Gr'g8013/v25 tionata vertical plate 0.29 b Ru 0 4 57xl0 2400 32 Transitional (a)Ru'04rSc) 1 ' 4 3xI040 0rScu2.5a10 7 L'width plate Cu 2< wragg68 and turbo- • (transitional) lentfree . . • convection at a horizon- (b)Nu.0.l6(GrSc)l/3 .2.5Z 107 (continued on p 47) - - a 47 • Table 1.3. (continued from page 46) a System Correlation Range Paremeters Electrolyte Authors 32 Turbulent 108tGrScil4alO12 l..length plate Cu2 Fenec h , Tobi as U free convec- 2100Scs52,000 Fe(CZi) Fanecha tion at a d0.i.0 for plate Boeffard4t' horizontal (d)Nu.0122(GrSc 9" 18 ) 11 4 surrounded by plate vertical walls; <1.0 for embedded electrodes l 33 Turbulent Nu0.57(GrSc)01' 2 . 9iGrRSC< 400 Rradiun Fe(CN) Ravoo7 free convec- 2S00 34 Free convec- 6xl0lsGrScc7,dO13 Nu.0.3(C GrSc)0.28 Lheight plate Cu2 Fouad,Alsned 75 tion at an Cf(angle of 1750 35 Free convec- Nu.A(GrSc) 032 22xlO • 36a Combined free bu1.96(ReSc d /1. 1.3a107 b same; no correlation Cu 2 wragg8 opposing flow (continued on page 50) 48 It is interesting.to notice that free convection on a rough surface (number 35 of Table 1.3) follows a higher GrSc dependence than for the same Ba range on a smooth plate. The roughness definitely appears to ease the transition to turbulence. Fouad and Ahmed 75 thvestigated free Convection at inclined plates, expressing the results in the form of a dependence on (CGrSc) 028 where • C = 1 for a vertical plate and equation (34) of Table 1.3 is obtained. The constant C corresponds roughly to cos 8, 0 being the angle of inclina- tion from the vertical, as long as the densification is stable, i.e., in • the direction of gravity. For unstable dénsificatiOn C increases sharply with 0, and least squares analysis gives a GrSc dependence which appears • to approach the exponent 1/3 as the plate becomes horizontal. Free convection ona horizontal plate has been studied extensively. Fenech and Tobias first established the (GrSc) 1 " 3 dependence of this kind of turbulent transport. The rate per unit area is independent of plate dimension, provided edge effects are unimportant. Another condition • investigated was the presence of an opposite parallel surface (a diaphragm) which did not influence the mass transfer unless closer than approximately • 0.5cm; An alterTlative equation for plates embedded in an inert surface was also given, which takes advection into account (eq. 35b of Table 1.3). At low Rayleigh numbers Wragg 60 found a lesser Ra dependence, more resembling that for laminar free convection. In this range a cellular flow pattern is believed to exist inana1ogy to thermal and surface- tension driven cellular convection (Bnard cells). In the region where • the convection is turbulent the Ra 1 " dependence has been confirmed over 7 decades of Ra by Ravoo, 79 who used a centrifuge to vary the body force 49 at constant composition As a general comment on free convection limiting currents it should be remarked that several experimenters'11 48 report difficulties in trying to obtain steady limiting current plateaus of good readability in those solutions where the driving force is small. In extreme cases curves with maxima and minima were obtained, and a limiting current could not be assigned. This aspect of free-convection mass transfer will lead us to investigate mass transfer transients more closely in a later chapter. Combinations of forced and free convective mass transfer (mixed convection) are amenable to theoretical analysis so long as the free convective flow is in the same direction as the forced flow (aiding flows). This is e.g., the case in free convection with forced flow in vertical channels. Ross and Wragg 93 have experimentally investigated such a situation and found a smooth transition from the forced convective Lv&ue correlation to the (GrSc) 1!4 dependence typical for laminar free convection. In opposing flows of this type a more complex behavior was found, 83 where the mass transfer rate is minimal at a certain critical combination of length, Reynolds number and Grashof number. Very likely at this point the leading edge of the diffusion layer moves from one end of the transfer section to the other. The flow instability becomes apparent in marked current oscillation and poor reproducibility. Hickman and Tobias 32 have been the only investigators to study co*bined free and forced convection in horizontal channel flow. They found a remarkably sharp separation of forced-convection dominated, as opposed to free-convection dominated,mass transfer s The critical Grashof number based on the diffusion layer thickness in forced convection,is • 50 • Table 1.3. (continued from p. 47) System Correlation Range Parameters Electrolyte iuthor 37 Combined free (a)Nu.0.85(ReSc d/L) t13 75 38a Flow at a Nud . l. 83Re ° Sc 1" d.diameter Cu2' Erishna. - freely rotating Re'aih/v cylinder Fe(CN) Jagannadharaju 40 cylinder in a h.distance baffled cylin- drical cell b Flow at ring Nud2.4lReOhZScl/l C.distance electrodes in C032 rotor- • the bottom •of. Re'vdh/-v bottom of cell same cell 39 Flow at a Nuw10.0.431ie2Sc1/3 8OO d'2.S3 cm 40 Flow at a • Nu'2v0.42(ReSc) 2 100 • 0.075cm 41 110w at a 0 • Nud . 4 lReSc 6OcReclO4 a=arnplitude Fc(CN) Somasundara, et al. 33 vertical Nukdfli l.ga.c' f'25sec d'dzaneter cylinder in • Re.fad/v 0.24ce transverse • 800sSc<22411 vibration (continued on p. 51) 51 Table 1.3. (continued from P. 50) System Correlatton Range Parameters . Electrolyte Authors 42a Floe at a Nu0.476Re56Sc3 lbS - 42k Floe at a 0 7 ffrequ ncy NudeO. 437Re Sc - vertical plate (W/T)0.S in vibration perpendicular Re2fad/s, to the thickness . 43a.. Floe at...... one ...... 79O. 38 ...... NUdO. 050Re 2000R0<4000 delength Ca2 2i wall of a Re/v Holrne5 • et al 2500 b Same in free Ne0.57(GrSc)uI4 S.4n109 44 Oscillating (a)Nud.o.24g09h25c1/3 lt 45 Annular cell Nu 0 243e 48 nztk tangen. (de lL) ''2 l.3 - rate 22 A.m(r-r.) ReQde/VA . flow entrance d .2(r -r.) e 1 (continued on p. 56) 52 Gr = gtthcdL > 920 (1.30) vu < If this number was smaller than 920, mass transfer could be represented by the forced convection correlation, otherwise by the.free convection correlation of Fenech and Tobias They also inferred the existence of cellularvortex flow near the electrode from deposition. patterns; the induction length for this deposition pattern agreed with the criterion (30). In studies such as these it is convenient and, as can .be proven experimentally, necessary for the accuracy of. limiting current measurements, that the electrode is divided in several insulated sections. A limiting current curve is obtained for each of these This technique, first introduced by Ferech and Tobias,U requires that the potential be kept the saae on the various sections as the limiting current is approached. Some complications associated with this condition will be discussed in 1ihapter.5. . . . . 1.8. Restricted cells with agitation Under this heading a number of investigations are grouped in which the electrolyte is stirred by movement of the electrode, or where this stirring (or circulation) is accomplished independently. . In principle the boundary with the laminar flows discUssed earlier is not sh .arp. Complications arise in the first place because of secondary flows in restricted cells. An example is the enclosed rotating disk mentioned above.. Furthermore boundary-layers on.the stirring electrodes may. become detached. Local turbulence can be generated near electrodes or inert stirrers Of interest is the mass-transfer to the stirrer, as * Numbers 38 through 45 in Table 1.3. * S 53 well as to selected points of the cell The first aspect has been investi gated more frequently, but with a refined technique the second could give valuable information about convection patterns in stirred tanks. Stirring by gas-evolution is considered separately in the next section An example of agitation by a free rotating cylinder is the work of Krishna and Jagannadharaju, 40 who measured mass transfer at a cylindrical rotor, as well as at.ring electrodes inserted concentrically in the bottom of the cell. An impeller attached to the free end of the rotor did not change the current. The clearance between rotor and ring- electrodes was varied. They found that mass transfer to the rings was maximal for those rings having the same diameter as the rotor (or impeller).: . Flow in stirred vessels was also investigated by Holmes a o 21 who simulated the.mass transfer in a diaphragm diffusion cell stirred by magnetic stirrer .bars. This is a good example of a model study with a direct practical purpose. A minimum stirring speedin such ôells is necessary to avoid appreciable errors in the cell constant. The experi- ment permits this stirring speed to be related to the solution properties. Mass transfer in a pulsating flow perpendicular to a stationary, plate 57 could be correlated by a laminar-type mass transfer expression at low frequencies, and a turbulent one at higher frequencies The pulsating flow s gcner.tec by a vibrating diaphragm, which forms the bottom of a cylindri- cal cell of 20 cm diameter, with an open upper end The mass transfer to the horizontal disk electrode is not dependent on distance to the diaphragm, if above 17.5 cm, nor on free liquid height 54 . Raju, Rao, and co-workers, 33 ' 71 also investigated mass transfer at vibrating electrodes, for low frequencies (higher frequencies would cause cavitation). Mass transfer follows a laminar type correlation for transverse vibration of a vertical cylinder, and for a vertical plate vibrating parallel to the face. The. Reynolds number in the case of the plate is based on the width, indicating predominance of form drag. For vibrations perpendicular to the thickness, skin friction predominates and the Reynolds number is preferably based on the equivalent diameter (total surface area divided by transverse perimeter).. Rotte and Noordsij 53 studied a vibratin.g sphere, i.e., a sphere. kept in oscillating motion by means of an attached rod held by a spring against an eccentric wheel. The results showe4 considerable spread and are approximately represented by a Pe 2 dependence, with a. constant term (due to pure diffusion) of correct magnitude. Exeriments with the same sphere in rotating motion showed less spread but a higher intercept, possibly due to secondary flow generated by free convection or temperature differences (cell dimension was only 6 times spherediameter). Local mass transfer measurements on rotating and vibrating spheres should be of interest from a hydrodynamác point of view.. . . . . An interesting modification of coaxial flow in annular cells was studieçl by Regner and Rousar, 86 who created swirling flow by means of a tangential inlet. The mass transfer correlation combines features of the boundary layer correl3tions (dependence on L 2 ) with those of turbulent flow between rotating cylinders (dependence on Re °78 ), although the Reynolds numbers do not reach the range where concentric flow is turbulent (29 or Table 1.3). Related is the flow in a hydrocyclone investigated by Ravoo 80 55 1.9. Cells with gas-evolution Electrolytic gas-evolution has been used as a means of generating convection 'in heat transfer experiments, 92 where the'primary purpose was to gain an understanding of the subcooled-boiling mechanism. Only more • recently has it been realized that electrochemical mass transfer itself can serve as a convenient model for heat transfer in the neighborhood of an "active," bubble-generating point. Weder 54 measured limiting • currents of ferricyanide reduction at concentric horizontal ring elec- • trodes surrounding a single orifice from which gas was bubbled. A depend- énce on Q030 (where Q is the gas volume flow rate) was 'found, in excellent agreement with earlier heat transfer results. Earlier, Venczel 14 had made measurements of limiting currents of ferric and ceric ions at a cathode where hydrogen was simultaneously • evolved. The bulk solution was titrated after theexperiment to establish the partial current due to hydrogen ion reduction. The electrode was oriented vertically. In this more complicated situation two effects can be expected to occur at the same time: convection ofthe solution due to the rising bubbles; and turbulent mixing near the cathode after detach- ment of a single bubble. The experimental dependence of mass transfer • • rate on v0525 (where V is the gas volume evolution rate per cm 2 electrode) • shows that the second effect is predominant. 1bl 102 proposed a modified "surface renewal" theory to explain this dependence, where 'at least on copper electrodes fair agreement was found with experiment. 56 Table 1 3 (continued from p 51) - System Correlation Range Paraaet.rs Electrolyte Authors 46 Floss at the Nudl4. e 3 &' 3G Re no - - L.lsngth Fe(CN) Ravoo8 • wail of a Re i /d d.dbmmeter • hydrocyclone G(d2htais6/d)2i(d/d.)l/6(tan6)4/3 cyclone • if 0>13 4 -diaanter • : G.0.14(d-0.46h/d) 213 (d/d) 6 inlet if O - l.2S - 41 Mass transfer c j 0_0 . O2 e ShSSc 2S 2 (l c )O 1 serficia1 Fe(CN) Jottrand, - 2/3 is • to a small velocity Grunchard • plate immersed ceporosity in fluidized dpaxticle bed dia.eter (or char.di.en- lion) 0.38 48a Mass transfer cJ 0 O.43Re ; . Fe(tN) Jagannadharaju Rao -. 45 to the wall in 200 - b Mass transfer jb..0 . 822ReP 38 -- • Fe(CN) Krishna 94 - • - to a screen • supporting a packed bed • (continued on p 57) 57 Table 1 3 (continued from p 56) Sy,teL Coriel.tjon Rgnge !araeters . Electrolyte Aothrs 49 Mas5 transfer to (a) 3S< Red 140 . d.lca d.sphere dieter Fe(CN; 3 a sLngle sphere Nu.1.44Re0S$sc1f3 Joll , Ifanratty 72 6Re<1500 soperficisl in a packed bed (b)ReI 40 ; 158 .... -- 58 * 1.10. Particulate flow systems. Recently the limiting current method has been increasingly used for studies in packed and fludized beds. In 1962 Jottrand and Grunchard' 8 reported on mass transfer t.o a small rectangular nickel plate inmersed in a liquid fluidized bed of sand particles. Mass transfer rates were S to 10 times those measured in an open pipe flow, and a maximum rate was measured at a bed porosity of 0.58. Vergnes, Le Goff a.o. showed later that this maximum is directly related to a maximum in the average kinetic energy of the fluidized particles per unit bed volume. 70 Measurements at the wall of fluidized beds were made by Krishna, Raju and Rao, 45 who used an annular cell, with varying radial ratios, to test the influence that an inner tube has on the fl.uidization. Only at low r/.r0 values is the inner wall coefficient appreciably higher than the outer wall coefficient, which is itself somewhat lower than in a pipe bed of the same radius. 55 Raju and Rao39 made measurements in an annular packed bed,with the inner wall as the measuring electrode. (anode). These measurements were later extended to both walls. 45 The correlation employs the conventional dimensionless quantities: k 2/3. = - Sc , and V vd Re = (l-c)v ... (1.32) Numbers 47 through 50 of Table .1.3. . . . 59 In these wall transfer.meãsurements stagnant liquid regions may play an important part (Kramers-Thoenes theory) It is interesting to observe the differences obtained by measurements in different places of the bed.. A beginning was already made by Krishna and Raju, 58 who measured at ring electrodes embedded in the perforated plates between which the packed bed was contained. They obtained virtually the same dependence: Re -0.38 (lj . 33) found earlier for the wall mass transfer 93 Ver.y refined measurements in various places of the bed were made. by Jolls and Hanratty, 72 who used an electrode sphere in a packed bed of 1 inch spheres. The over-all mass transfer data suggest a dependence -0.42 -0.44 Red (turbulent) or Red . (L34) outside the low laminar flow region (Re < 35), where the flow rate dependence is uncertain. A transition to turbulence occurred between RellO and 150, but no conspicuous change in the mass transfer ensues because largefiow fluctuations occur over the whole range. . The rate expressions 49 of Table 1 3 are valid only for a single active sphere in a bed of inert spheres. In a bed of active.spheres the spheres are bathed in the detached difusion layers of other spheres and the rates 3re lower.. This is, of course, a general objection to the study of single activated elements of a particulate system Nevertheless, such a study can yield valuable information about flow patterns, and variations throughout the system 60 Hicks and Mandersloot62 investigated flow systems with turbulence • promoters, where the orientation of the promoters (or packed iEregular particles) is an important parameter. In such systems the Reynolds • number d G/(1-E)1i. used. for packed beds of spheroid particles is not effective. Instead, they correlated. their data with the "viscous part" of the pre .ssure drop, based on the consideration that the average transfer rate in turbulent flow over a flat plate is proportional.to the square root of the shear stress (analogous to expression 2 of Table 1.5). The modified Reynolds numbertherefore is based on the .voi.d velocity VIE, the kinematic viscosity v, and an equivalent diameter s/a where sis total area per unit volume and a adimensjóna]. coefficient derived from pressure drop data: AP - —2 = apv + bpv Then: .. . . Rem aev . . . (1.36) and the expected mass transfer correlation is: . -1/2 = const Rem . (1.37) .This expectation is very satisfactorily borne out by limiting current measurements. . . The application of a fluidized bed of conducting particles as a statistically continuous electrode has been discussed by Le (off a.o. 70 Interesting similarities with heat transfer in fluidized beds could be exploited in this way. . . . . . 61 1.11 Electrochemiluminescence An interesting variant of the usual limiting current method is the 43,44 technique of electrochemilum inescence, where a blue glow appears on the anode when a potential is applied between platinum electrodes immersed in a solution containing H 2 0 2 and luminol* in very small amounts. The intensity of the glow can be measured by a photomultiplier and is, in laminar flows, proportional to the mass transfer rate of H 2 0 2 to the anode, where it activates the reaction. The proportionality constant has to be derived from calibration in known transfer geometries. Likewise for the correlation of mass transfer rates the diffusivity of 1I2O2 is necessary and was obtained by Springer a.o. 44 from measurements of luminescence on a flat plate anode in laminar flow. The technique appears to be especially suitable for flow visualization. 43 1.12. Conclusions it is clear from the enumerative review presented here that the limiting current method has established itself as a powerful tool in the study of, mass transfer rates in chemical engineering operations. In the ease and accuracy with which currents at electrodes can be measured it compares favorably with heat transfer techniques used to establish rate correlations One should realize, however, that the con- ditions during the measurement are different Heat transfer is most readily measured with a uniform flux generated at the surface In electro- chemical measurements also it is very easy to impose a constant current at the electrode. There is, however, no assurance that the current distribution will be uniform unless it is imposed (by means of separate, insulated sections) The interplay of. • concentration overpotential, kinetic effects and.ohmic potential drop between the electrodes can give rise to complicated current distributions Even if a uniform distribution is externally imposed, the ovérpotential measured at the electrode is not simply related to the concentration at the electrode Only in the limiting current condition, where the over- potential is predominantly concentration overpotential, does one have assurance that, at the electrode surface everywhere, the concentration is • zero, or effectively zero. The limiting current method in mass transfer and the usual heat transfer techniques are therefore in a sense complementary, since they correspond to different boundary conditions. Over the most popular non-electrochemical technique of measuring mass • transfer, by dissolution, the limiting current method has the advantage of keeping the transfer surface unchanged, at least in the redox reactions This is of great importance, e.g., in work on tUrbulent mass transfer. Furthermore, instantaneous flux measurement is possible so that various flow conditions can be tested in one experiment. On the other hand, the method is restricted to electrolytic solutions, which have a high Schmidt number. Mass transfer in gases, e g , differs significantly from that in electrolytic solutions, in particular for tur- bulent flow, and is closely analogous to heat transfer. However, many mass transfer situations are more readily analyzed for the condition of a high Schmidt number, so thatthe limiting current method has a considerable value for comparison of theory and experiment. 63 It is of interest also to conipare optical techniques, in particular interferometry at an electrode, with the limiting current method. Measure- ment of interference fringes allows us to observe mass transport below the limiting current. Also, since it directly visualizes the diffusion layer, it is helpful in correlating theory and experiment (as, e g , in the turbulent mass transfer workof Lin a.o. 4 ), and in establishing the onset of. secondary convection.patterns, boundary layer separation, etc. However, the disadvantages are considerable: For a cell of finite thickness refraction of the beam (into regions of higher refractive index) causes serious difficulties of interpretation. Thin cells, on the other hand, influence profoundly the flow profile Interference fringes are crowded into the region of greatest interest, near the electrode. The reading of the slope at the electrode becomes almost impossible at high current densities, e.g , at or near the limiting current, where this would be of great interest Even the position of the electrode surface on the interferogram depends on the setting of the focal plane. 115 3 A parallel plate channel geometry is practically the only con- figuration that the method allows There is no question of the more general applicability of the limiting current technique Interferometry may serve in discrete instances as complementary to the limiting current technique This review of experimental work has brought to light several aspects of the method, that are in need of further clarification. i 64 1.. The use of correct and precise diffusivities is decisive for • its accuracy if one wants to derive mass transfer correlations. In particular for the copper deposition reaction accurate diffusivities are not available over the whole range of Cu 2 -concentrations. The diaphragm- cell diffusivities used in free-convection work 11 are significantly higher than the effective diffusivities for solutions dilute in Cu 2 , as measured by Arvia 17 and successfully used by Wragg 63 to correlate laminar mass transfer in annular flow. At electrodes where the current-distribution at the limiting current is non-uniform, the limiting' current plateau is often of a poor quality. , It appears th.at , in such cases, sectioning the cathode and •reading separate sectional limiting currents markedly improves the quality of the curves. It is desirable to understand whether this is a typical characteristic of extended electrodes, or somehow connected with the, particular system (Cu 2 ) and geometry (parallel plate channel) used. In other words, what factors determining the shape of a limiting current curve can be controlled, and which other ones are inherent in the reaction, or the geometry? . . . . . . • In free-convection experiments with very dilute solutions, the attainment of steady state is so slow that often.no definite limIting current can be read. If, on the other hand, the limit±ng current is approached too slowly surface roughness, or depletion of the bulk solution, may create an unsteady state ofa different kind. What are reasonable' approach times to be used in forced and free convection limiting current experiments? • . . . • . . . 66 Chapter 2 VALIDITY OF THE LIMITING CURRENT METHOD Part I Basic concepts of electrochemical mass transfer. 2.1. IntrOduction.'.. In the first part of this chapter the concepts of migration, electrode polarization, and current distribution are briefly reviewed since they are, at least as far as their quantitative aspect is concerned, less familiar to chemical engineers than to electrochemists. More detailed expositions of the fundamentals of electrochemical mass transfer, from a physico-mathe- 103,109 matical viewpoint, can be found in the recent literature The material brought together in the next sections is aimed at a quantitative understanding of what factors determine, in the steady state, the limiting current level and in what sense the mass transfer process, in that condition, is different from that bel.ow the, limiting. current. The electric field is found to influence the limiting current level and this efféct.can be expressed in terms of electrolyte composition. The results are relevant to the inter- pretation of limiting currents, i e , in particular, to the diffusivities used in the correlation of mass transfer rates calculated from limiting currents. (Chapter 3) However, the electric field also is the cause of dissimilarity between the current distribution at the limiting current and that prevailing below the. limiting current. Some results discussed in Chapter 4 can be understood in terms of this dissimilarity, interacting' with unsteady-state mass transfer. . . . . . .. . . . The concepts discussed in the first part of this chapter allow us to • define an ideal limiting current measurement or, rather, to perceive what inherent limitations of a given electrode-reaction cause it to produce a 67 poorly defined limiting current plateau. Part 2 of this chapter defines S in a more general way the conditions for valid measurement and interpre- tation of limiting currents, taking into account that the measurement is • by nature an unsteady-state phenomenon. 2.2. Migration and diffusion When a potential is applied between conducting surfaces submerged • in an electrolytic solution, a surface reaction will take place at each electrode. A general expression for this electrode reaction, which may be a deposition or dissolution reaction, or a redox reaction involving only species in the solution, is: . . s.M.Zi ne . (2.1) The current density by stoichiometryisproportional.to the reacting ion flux (or fluxes, in the case of a redox reaction): s.i = -nFN. • . (2.2) If the current measured serves to evaluate the mass transfer rate due to convective diffusion, it is desirable that the electric field make a • • negligible contribution to the flux of reactant species at the electrode. Since the electric field can never be completely suppressed, it would be • preferable to know its contribution to the flux exactly, so that it could be accounted for. • The migration contribution could be determined experimentally. for .any. combination of reacting and inert ions, if the reacting ion diffusivity (DR) • 68 would remain constant upon addition of varying amounts of supporting electrolyte. The migration contribution to the current would then simply be: 'L'D where i is the measured limiting current and i the limiting diffusion • current, i.e., the limiting current measured with the same reactant con- • centration but enough supporting electrolyte to suppress the electric field completely. In the acidified cupric sulfate solution this condition is: • r = cHSO/(SO )j . (2.3) 2 4 2 4 CuSO4 Although this condition can obviously not be realiz, the value of could be extrapolated from the beh3vior of 1L/1D near r=l. However, in the first place the behavior of ionic diffusivities with concentration variation of the supporting electrolyte is a priori unknown. • Furthermore, the migration contribution would not necessarily be the same in various flow geometries. It is, therefore, important to gain whatever insight we can from theoretical calculations based on assumptions that are not unduly restrictive. For dilute solutions the contributions to the.flux of the ionic species i are: N. = -D.Vc. - c.z.u.FV + yc. 2.4) • •— . ( i . The second termrepresents the migration in a field of strengthE=.-V$ . The mobility u 1 is the .average velocity due to a unit force per mole 70 (e.g., expressed in cm 2mole/Jsec). It is related to the ionic diffusivity .D. by the Nernst-Einstein equation: D. s . RTu , (2.5) valid in the limit of infinite dilution. In the limiting current method the reacting ion concentration at the electrode is made vanishingly small by applying a large negative,.or positive, potential. Even though, according to (4) the direct migration contribution to the reactant flux at the electrode vanishes (except in the case of a binary electrolyte), there will be an indirect contributián of the electric field because it affects the concentration profiles of reacting and inert ions. The species conservation equations: c. 1 • = - V•N. • (2.6) • are interdependent because electroneutrality. S: preserved everywhere in • • the solution, except in a very thin double charge layer near the electrodes. Therefore, • • • • • z 1 c. = 0 . • (2.7) • • • Since the current density in the solution is defined as: • • i = FEZ . N. (2.8) • • the e].ectroneutrality condition implies that: V•i= 0 . (2.9) 71 In the bulk solution the net transport of charge takes place by migration only. Near the electrode concentration gradients develop and diffusive transport occurs. For each inert ion•convective diffusion and convective migration cancel each other in the diffusion layer, since the total flux toward the electrode must be zero according to (2).. For the reacting ion, on the other hand, a shift occurs in the relative contribu- tionsof diffusion and migration to the reacting•.ion flux, which.is constant throughout the diffusion layer. Figure 2.1 shows this schematical-. ly for the system: Cu (reacting cathodically), H', SO . A common diffusion layer thickness has been assumed for all three ions In reality H, having the highest mobility and diffusivity, spreads out farther into the bulk solution than Cu and SO In certain cases of free convection, namely, when the density difference at the limiting current is relatively small, e g , in the ferricyanide-ferrocyanide redox reaction, a large excess of highly mobile supporting ion, in this case 0H, tends to make the velocity profile unstable.. 113 The diffusivities and mobilities of the inert ions influence the concentration profilesdirectly and indirectly. The migration flux in (4)is proportional to the ionic mobility and concentration, but alsoto the field strength and, therefore, indirectly dependent on the total concentration and the mobilities of the other ions Thus the migration contribution in acidified cupric sulfate solution can be expected to be lower if H2SO4 is completely dissociated to SO and two highly mobile H, than if the dissociation is only partial, with one ion present for each HSO ion 72 For a binary solution, i.e., an unsupported solution (r=0), the migration contribution can be calculated directly if the ionic mobilities and diffusivj.ties of thetwo ions are known. In such a solution the equations (6) for both species can be combined to one equation: -D V 2c + vVc = 0 , (2.10) where D and c now refer to the molecular diffusivity and c,oncentration. Equation 10 is the analogue of the equation -D.V2c. + = 0 , (2.11) for the reacting species i in the presence of an excess of supporting electrolyte, such that the electric field is virtually suppressed. In dilute solutions the binary diffusivity D 5 is related to the ionic dif- fusivities by z u D + u 0 (z -z )D 0 ++- .+ + - + zu -zu zD -zD 212 • ++ - - ++ -- One can now express.the limiting current 'I. as a multiple of the limiting diffusion current iD;.for laminar forced convection i is proportional to • D1/3 so that in a cupric sulfate solution, e.g., • D 213 =s • . .. i /i = 1.883 R R based on the mobility ratio u 0 -'u ++ = .1.493. 4 Cu • From (12) it is evident that binary diffusivities are, in principle, not appropriate for the correlation of mass transfer rates calculated from limiting currents. The binary diffusivity of CuSO4 , e.g., is approximately • •••• • • : 73 v• .#. 04 03 02 + 01 9' H. + U 0I0 °c'j —0I —02 —03 —04 75 20% higher than the ionic diffusivity of Cu (both at infinite dilution); this is due to the higher mobility of the SO ion, compared to Cu. • For solutions with composition intermediate between a binary solution • (r=O) and a well-supported solution (r=l), the migration contribution has • to be calculated numerically. The equations of convective diffusion for • all species: . • . . V . N. -D.V2c. - Z1u.FV (c1V) + v•Vc. = 0 (2.14) are solved together with the equation of electroneutrality (7) and appropriate boundary conditions. The results of interest are the ratio and the .inert ion concentrations at the electrode, as a function of bulk composition: varying from r=O to r=l. For a specific ternary ion combination the parameters appearing in the problem are the ratios of the inert ion mobilities to the reacting ion mobility, and similar ratios for the diffusivities. Ifthe Nernst-Einstein equation (5) is applied to the extent that diffusivity ratios will be equal to mobility ratios, also in more concentrated solutions, the only two parameters (mobility ratios) remain.. This constant-ratio assumption is plausible for solu- tions dilute in the reacting ion but concentrated in supporting electrolyte. • The more complicated case where the reacting ion concentration is also • • high, is not properly represented anyway by the constant-property equations (14). - • • • For a number of frequently used ionic combinations Newman 123 and Co- • workers 3 have solved the migration equations (14)., using infinite-dilution • mobility ratios as parameters. Figure 2.2 shows the function 1L'D as •a function of r(3) for the acidified cupric sulfate solution, assuming 76 either complete dissociation to SO or partial dissociation to HSO only. The calculations were made for three different kinds of diffusion layer: laminar forced convective; laminar free convective; and penetration.-. type diffusion layers. The last type includes time-dependent pure diffusion in a capillary cell and at a growing mercury drop. Laminar forced convec- tive diffusion layers include those at a rotating disk and in Lvque- type mass transfer. Figure 2.2 shows that the migration effect in free convectjon.as r 1 is anomalous; this has been discussed elsewhere. 13 The most conspicuous feature of these results is the dependence of the migration contribution on the degree of dissociation of the HSO ion, rather than on the kind of diffusion layer. This is also shown strikingly by Figure 2 3, which presents the solution composition at the electrode in terms of the relative acid accumulation: . (CHSOthd - cH SO bulk )/Cc blk . . (2.15) The composition of the solution near the electrode has long been of interest to investigators of the copper electrorefining process, which . j traditionally conducted in free convection. By pinhole sampling 68 and by freezing 126 of the diffusion layer it was found that H 2SO4 is present in the cathode layer in concentrations exceeding that of the bulk by approximately 0.5 to 0.7 times thebulkCuSO 4 concentration. From Figure 2.3. one can see that this would correspond to complete dissociation of HS0. This.appears to be at variance with the rather small dissociation constant (0.0103. at 25 ° C) reported for bisulfate ion in very dilute solutions. Recently, however, Flsueh and Newman.125 have 77 employed results of Raman spectrum measureinents.jn aqueous H2 SO4 solutions to calculate the stoichiometric dissociation constant of HSO ; they found an increase by three orders of magnitude as the acid concentration increases from very dilute to 3 molar. Optical measurements at a capillary cell cathode 125 cOnfirmed the high values for the relative accumulation indicated by earlier, primitive methods. The evidence, however, is not yet conclusive. An accurate determination of the composition of the solution at the cathode is especially desirable in free convection since the driving force, • the density difference with respect to the bulk, is. affected strongly by the degree of dissociation. Also the excess current due to migration • increases sharply with a small deviation from complete dissociation. This will be of importance in the interpretation of rotating disk integral diffusivitjes reported in Chapter 3, since these diffusivities include a migration effect. An accurate assessment of the effect will • be attempted by computation along the lines indicated by Newman and Hsueh. . • • • • 2.3.Polarjzatjon• The limiting current reveals itself as a plateau or inflection • point in the plot of current vs. electrode potential; The potential of the electrode is measured most conveniently as the "overpotential" with • respect to an identical reference electrode, i.e., apiece of the same metal as used for. the working electrode, immersed in the bulk solution. (Figure 3.2). An arrangement which uses a liquid bridge, is quite common where one does not want to disturb the flow in the main cell. In. 78 case the essential condition is that a negligible current is flowing between the reference electrode and the working electrode. The potential measured between these electrodes can be thought to represent the sum of two cosnponents. One of these would be measured between the reference-electrode and a second, imaginary, reference- electrode in the liquid adjacent to the working electrode (but just outside the double layer). The second component is then the potential difference between the imaginary reference electrode and the working electrode; following Newman, 124 it will be called the surface overpotential (fl5 ). It is dependent on the reaction rate and the concentrations at the electrode. It can also be identified as the potential drop over the double layer at the electrode. The first component of the overpotential is interpreted, in the most general approach, in terms of electrochemical potentials of the ions in equilibrium with the reference electrode. The potential of a movable reference electrode, with respect to a fixed one, is _flFVEr =sV (2.16) where 11i is the electtocheniical potential of species i. For dilutesolutions this can be written: s.T (217) where -VO is the electrostatic field strength in the solution In the • absence of concentration gra.dients two reference electrodes measure the ohmic potential drop in the solution, since 79 S. V~P = - + . Ezvc , (2.18) where: 2E 2 • K = F Z .U.C. . (2.19) • . iii 1 If no current is passed an electric field may still exist if there are concentration gradients; this so-called diffusion potential is caused by unequal ionic diffusivities. One generally distinguishes three contributions to the overpotential, besides the surface overp.otential: an ohmic potential drop; a diffusion potential, caused by the concentration gradients in the diffusion layer, and a !'Nernst" potential difference which is identical with the thermo- dynamic EMF of a concentration cell having the two reference electrodes as poles: RT c = i £n - . (2.20) conc.celi nF C 2 In reality the EMP of the concentration cell is lower by the magnitude of the diffusion potential (Corresponding to the classical "cell with transference") Two half-cells cannot be combined without creating a concentration gradient which gives rise to an electric field opposing the over-all potential gradient For this reason the diffusion potential and the thermodynamic potential are jointly designated as concentration over- potential. (nc). The complete expression for the concentration 80 overpotential includes an ohmic contribution due to the variation of conductivity in the diffusion layer: 1c y=of FJ' .]dY . io (2.21) where subscripts 0 and b. stand for, respectively, electrode surface and bulk. • This expression can be simplified considerably for the case of a binary electrolyte and for a minor reacting species in excess supporting e1ectr.olyte. For, metal deposition from a binary electrolyte the concentra - tion overpotential reduces, after application of (4), to: = R0T1 Ui— - . _)fLn . - t(l- (2.22) • • The sèond term in the brackets is not negligible except at the limiting current the concentration overpotential in •a symmetric .binary electrolyte is twice that obtained with excess supporting electrolyte. In the case of a minor reacting species with excess supporting electrolyte, the conductivity of the solution in the diffusion layer makes the first and third term in (21) negligibly small compared to the thermodynamic overpoténtial so that: s1RT in C ib ~ : (2.23) • nc Summarizing., the overpotential measured with respect to an identical reference electrode may be represented as the sum of three contributions: 81 n=fl s .+nc.+ (2.24) The ohmic contribution: r -,. dr (225) K b of equation 2.30, depends on the position (r) of the reference electrode. In a situation where the other contributions are absent, it can be cal- culated by solving Laplace's equation for the geometry in question, with cathode and anode being equipotential surfaces Characteristically the ohmic drop depends only on the geometric parameters of the cell con- figuration, and., of course, on the conductivity of the solution. For example, for a disk electrode embedded in an .inert plane the ohmic drop from an equipotential disk to an infinitely far removed counter- electrode is given by . . . irir IR = .= . total . 4r 4K . (2.26) 0 Figure 2.7 shows •the. equipotential lines and current flow lines near a disk electrode One notes from this that the placement of a reference electrode tip near the edge of the inert disk, where it cannot disturb the flow toward the disk (in rotation), includes almost 90% of the ohmic drop between cathode and anode. If the reference electrode is placed in the plane of the disk, at a distance r, the e.xact value is: 2 2tan!-. - 1) 1 / 2 AO r . . . = . . . . (2.27) total .. .. . 82 • To exclude the ohmic drop from the overpotential measured by the reference electrode, one resorts to the use of backside-capillaries, i.e., capillaries connected from behind to an orifice in the electrode surface, • small enough not to disturb the boundary layer at the electrode. The orifice should have insulated walls, since otherwise it would act as a pore electrode. If a suitable expression is available for the surface overpotential, one can construct a theoretical polarization curve, i.e., a limiting • current curve. Since the concentration overpotential as well as the surface overpotential depends on the reactant concentration at the elec- trode, this concentration has tobe assumed. Areasonable assumption is that the Nernst diffusion layer thickness is not very dependent on the current level. Then one can write for cathodic currents: • • :. (2.28) • • Cb If constant physical properties are assumed, this expression is obviously valid. A theoretical test assuming variable physical properties has, so far, only been made for a. binary electrolytic solution•(CuSO 4 ) at a rotating disk. Hsueh and Newman 98 showed that (28) is a good approxima- tion; the largest deviation, occurring at = 0.5 is 6-8% for 0.1 - 0.5 M CuSO4 . • • The surface overpotential n 5 can be related to the current density and the surface concentration by the Volmer-Butler expression: i = j {exp(aflZF/RT) -exp(-8flZF/RT)} • (2.29) 83 where Jo is the concentration-dependent exchange current density, a and are anodic and cathodic transfer coefficients. Z is the electron transfer number: • 2 = -zz/(z_z) for a binary solution; (2.29a) Z = -n for a solution with excess supporting electrolyte. (2.29b) • Equation (29) is here used as a functional relation, without regard to its implications for the reactionmechanism. One may consider j0 dependent on a power of the reactant concentration: j i(c/cb) Y , (2,30) where i is taken at the bulk concentration. Equation 29 is then an expression for n with four adjustable parameters. At large negative • potentials only i, and y are effective. The steady-state relation between i and r can now be obtained by solving the set of equations 21-30 for a given electrode reaction in a given electrode configuration, where i is known. • To write these equa- tions in a concise form adopt the definitions: • £ = , = . (2.31) Then: = + + , and • (2.32) c £n(l-f) + tf • (2.33) (with excess supporting electrolyte t is zero.). 85 Figure 4a shows this (1-n)curve for a solution of 0.1 M CuSO 4 at 300 rpm, and the same if 1 M H2SO4 is added as supporting electrolyte In the latter case the ohmic contribution is negligible (N' z 1) For the reactions most frequently used in limiting current studies the surface over potential is not necessarily small For the reduction of ferricyanide at a nickel or platinum electrode the surface overpotential is indeed very smali 167At a Pt electrode in excess KC1 solution the rate constant: 1 (2.40) cbnF has values of the order of 10_i cm/sec 10, the "standard exchange current density," i.e., i0 for a (hypothetical) 1 N solution, is 10A/cm2 . However, for the copper deposition reaction the surface ovérpotential is not negligible. Mattson and Bockris 134 found values of i 0 of the order of 1-10 ma/cm in highly purified solutions of CuSO4 in 0.5 M H2SO4 (see Figure 2.5). It is likely that in the so•lutions used for mass- transfer experiments, which because of their bulk cannot easily be kept free of organic contaminants, i0 is even lower. The complete equations (32-34) including surface overpotential can be solved by means of a Newton-Raphson root finding iteration for given values of f The program is reproduced in Appendix A. Figure 4b shows the results for the 0.1 N CuSO4 solution at 300 rpm, with and without supporting electrolyte The kinetic parameters adopted are those of Mattson and Bockris 134 = 1 50, az = 0.50 y = 0 42,* i0 (at 0.1 M CuSO4 ) = 4 2 ma/cm 2 IC 'AL 0.5 L) I 2 3 88 If the ohmic contribution is negligible, as with excess supporting electrolyte, the (f-n) curve depends only on M and the exponential coef- ficients a, B, y. Figure 6 shows the dependence on M for the same a, B, y values used in Figure 4. The value of n at = 0.5 shifts to.more • negative values as the exchange current density decreases. • In accordance with Tafel's law the shift for every decade increase in M has a constant . value if ui>>rn i (Mo > -10). At 25 ° C: f0.5 = - 25.679(n10) = -118.3 thy . (2.41) The value of this "half-limiting potential" is given, for large ii n > 90 my) by • • • -n = 29.58. {log M - 0.3010 (l-y-B)} • • (2.42) The assumption that the ohmic contribution is.negligible reduces to the criterion • • • ri << • log M • • • • (2.43) • • For a disk of r 0 = 0.2 cm and a solution of more than 0.5 M H 2SO4 strength, this is equivalent to i << 400 ma/cm 2 . The supported solution of Figure 2.4 has = 46 ma/cm2 at 300 rpm. FOr more dilute solutions the criteriOn 43 is certainly fulfilled. . . . . . . Therefore, equation 42 might be used to estimate .kinetic parameters from the measured limiting current curves. But (42) contains.three such 90 parameters; therefore more information is needed. Significant for the r.elative magnitude of the exponential coefficients, and y, is the slope df/do of the curve at the "half-limiting" potential: • fdf\ \Jf=0.5 2(1+y+8) 2.44• • The third relation between a and y could be derived by making certain assumptions about the reaction mechanism. Mattson and Bockris, 134 e.g., proposed that, at higher currentdensitjes than those corresponding to the linear polarization region, the mechanism of the copper deposition • reaction is • ++ slow + Cu + e Cu • (2.45a) +. fast • Cu + e. -.-, Cu , (2.45b) • • • and the reverse for dissolution. In that case would very likely be 0.25, a value found by the authors in extremely pure solutions and by many other investigators in solutions of various degrees of purification. For y Mattson and Bockris 134 deduced the relationship I (2.46) • However, from experiments in extremely pure solutions y is found to be approximately 0.4 (Figure 2.5). In solutions which are not carefully purified, such as commonly used in limiting current studies, organic contaminants occur. These may decrease the exchange current densities considerably, as can be inferred from studies by Karasyk and Linford. 170 • . . 91 Therefore, it appears advisable to assume = 0.25 and to derive values of y and M from equations 42 and 44. This is the course adopted in the next chapter, where steady state limiting currents for various rotation rates need.to be constructed. From.a curve for a relatively low current level,where surface roughness effects are presumably not very important, values of y and M can be derived which serve to construct an "ideal" limiting current curve for high current levels. 2.4. Current and potential distribution. . . The general definition, of the overpotential with respect to a reference electrode can be applied to the anode as well as to the cathode. If these two overpotentials, and the ohmic potential drop between cathode and anode (for uniform, i.e.;, bulk, concentration), are added together, the applied cell potential is obtained: . = cathodé + anode + Ablk . . (2.47) In the calculations of Figure 2.4 the reference electrode was taken to be the anode. This implies that the dependence of overpoteniai on current density of the anode is negligible, which 'is.usually ensured by making the anode area 100 - 1000. times as large as the cathode area. In any electrode configuration where cathode and anode are of equal area, e.g., in the important case of parallel plate electrodes, the limiting current should be measured against a reference electrode. If the. applied voltage were taken as the potential. indicator, the plateau. would be obscurred by the nonsimultaneous occurrence of diffusion . limitation at the anode as well as the cathode If also convection and mass transfer are interdependent, as in free convection between parallel plates, this can lead to problems which are best avoided by separating anolyte and catholyte by means of a diaphragm. In equimolar solutions of fern- and ferrocyanide between parallel electrodes the limiting current is reached first on the anodic side since the diffusivity of ferrocyanideis less thanthat of ferricyanidé 46 (see Table 1.4). Boeffard h3s performed calculations to obtain the minimum ferrocyanideconcentration, at a given ferricyanide concentration, necessary to obtain equal limiting currents at both electrodes in free convection. Apart from the complications caused by the existence of anodic current limitations, there is good reason to question whether it is possible to reach a virtual zero concentration at every point of the cathode simultaneously. This will be considered in detail in chapter: 4. In order to clarify what occurs during a slow (quasi-static) passage to the limiting current, we discuss here what factors determine the current distribution below the limiting current and to what degree it is different from that at the limiting current In the limiting current condition, i e , with c 0 virtually zero everywhere, the current distribution is characterized by the predominance of concentration overpotential Practically the entire applied voltage is used up in maintaining the concentration gradient at the electrode The ohmic drop bulk is unimportant and, consequently, so is the location of the counter-electrode The concentration gradient is determined by the hydrodynamics of the electrode flow cell. 95 Thus, certain limiting current distributions are uniform, e g , at a rotating disk of infinite extent; most, however, are nonuniform, e.g., the Lvque-type distribution for developing mass transfer where the velocity profile is linear. In the absence of surface and concentration overpctential the current distribution is determined exclusively by the geometrical characteristics of the electrode configuration. In the absence of concentration gradients the conductivity is uniform everywhere and the Laplace equation describes the electric field 0 , (248) with the boundary conditions = constant at electrode surfaces (2 49a) a. = KV = 0 at insulating surface (2 49b) Figure 2.7 shows current and potential lines between a disk electrode embedded in an insulating surface, and a hemispherical counter-electrode at infinite distance, for such a hypothetical primary current distribution Characteristic for the primary current distribution is the infinite current density at the edge of an embedded electrode (unless the insulating wall is perpendicular to the electrode) Since uniform current density at a rotating disk is reached, only limiting current the at the limiting current b one can expect that below the/non-uniformity caused by the electric field will persist to some extent Also, one can expect that at the limiting current the potential drop AO, from a reference-electrode to various points near the disk but • •. 96 1.8 F.. 16- - 14- - 0. I2- - c o 4K cx,ro/I for • 0 1.0 uniform current density • 08 - - >• 5, 0 06 L/( avg for. - uniform potentiai 0 4 02- - 'I • 0 0 02 04 06 08 10 r/r 0 M U B 10281 Fig 2 8 Current distribution at an equipotential disk (primary current distribution), and potential distribution at a disk with uniform (limiting) current density. From Ref 135 97 just outside the diffUsion layer, will be non-uniform. The highest bulk solution potential will exist at the center of the disk,. in order to maintain there a current density that is two times as large as that at an equipotential disk for the same total current. Figure 2.8, which illustrates this, is taken from Newnian, 135 who calculated the potential difference between the bulk solution at the center and at the edge of the disk: .rjiJ A= 0.3634 o L volt , (2.50) or, expressed in the dimensionless potential (31). 0.3635 NO = 0.3635 Q,1im (2.51) The connection with the parameter N' (38) implies that the potential variation across the disk is, at the limiting current, a fixed fraction of the ohmic drop between electrodes, where the latter is taken to mean: between the center of the disk and the equipotential counter-electrode. The dimensionless quantity NO obviously plays an important role in characterizing the non-uniform current distri.bution below the limiting current, where concentration and surface overpotential are not negligible. A similar quantity can be defined for other geometries. In particular, for aplane electrode of length L . . . . LI 1 LIZF . . . NO = . . . (2.52) Kb Figure 2.10 shows the primary and limiting current distributions 100 for such a plate, with the counter-electrode at distance h (Figure 2.9). If the distance h is small the primary current distribution depends on 4 the parameter h/L; it is more uniform at small h, the edge effect being confined to areas of approximate length h/2. At low current levels, where concentration polarization is not important., the current distribution is not identical with the primary dis- tribution because the very high edge currents automatically cause a non- negligible surface overpotential, even if no concentration gradients would develop at the edges. The so-called secondary current distribution corresponds to this combination of kinetic resistance and ohmic resistance of the electrolyte, in the absence of concentration gradients. Secondary current distributions have been calculated mostly for two simplificatioris of (29). One is linear polarization: i = i O ( + ) !TnS . . ( 2.53) valid for lit >> lo , the other Tafel polarization - RT i) , (2.54) valid for Jij >> i. . . . . . . .. . The expression (53); used inste of (49a) as a boundary condition for solving the Laplace equ4tion, intrQ4uces the dimensionless pararneter .iLZF . p (2.55) bo for a planar electrode The current distribution (i/iavg) is not dependent on the current level, only on (c+8)J The case J - ° o corresponds to the primary current distribution - 101 At higher currents the current level becomes. important. If (54) can be applied, the value of i0 is irrelevant in the solution of Laplace's equation. The current distribution, 1/'l avg is determined completely by: alilZFL or fN (2.56) Rob TK These secondary current distributions can be calculated by classical analytic techniques only in the most simple configurations. Numerical solution is often more convenient. Obviously a solution including the effect of concentration over- potential presents even greater difficulties, Newman 135 has provided such a solution for the rotating disk electrode, and (with Parrish) 136,137 for plane plate electrodes with parallel.flow., In the latter case solu- tions were obtained only for distances h > 0.5 L. Figures 2.11a and b show. the current distributions at a planar , cathode at four intermediate values of the average current level, according to Newman and Parrish. 137 The parameters employed are: J(55); N, which related to N' (52.) by . . N' = 0.5384 N . (2.57) The transference number, t, as well ascz, , and y (equation 29) which have been given arbitrary values. The parameter J is 0, indicating Tafel polarization. . . . . . . Avery important conclusion to be drawn from Figure 2.11a is that the Current distribution approaches the limiting, current distribution only at current levels very close to the limiting current. The degree to which 5 2 E 0 104 the current distribution retains non_liziting character close to the limiting current, depends on the parameter N and on h/L. For high values of N and h/L the current distribution exhibits a local maximum near the trailing edge, at current levels above. 50% of the limiting current. Figure 2.11b helps to explain the origin of these local maxima. At low current levels the current density at the edge tends to be high; it may exceed the eventual limiting current. At higher current levels the reactant supply at the electrode is a limiting factor and the increase of overpotential toward the edge. does not increase the current density any further. The reactant concentration at the electrode has reached a virtual zero level near the trailing end of the plate, while the average current is only 95% of the limiting current. . . It should be noted, however, that these calculated distributions are steady-state distributions. . A slow enough transition from one inter- mediate current level to the next, should still lead to a simultaneous limiting current condition, i.e., a uniform overpotential along the plate. How slow this transition has to be, one cannot infer from the information presented here. Earlier experiments in this laboratory have shown that in copper deposition on planar electrodes the limiting current is invariably reached first at an edge: at both edges in free convection (Fenech;hla increase of applied potential), and at the trailing edge in forced convection .(Hickman; 56 increase of applied current). This was first established by observing the hydrogen evolutionon the electrode; later, also by sectional limiting current measurements. The nonsimultaneous limiting current condition along the electrode obscured the plateau if the total current was being 105 recorded; consequently, it sectioned electrode with separate reference electrodes for each segment has been • used in later work. In Chapter 4 the rationale for adopting such a painstaking procedure will be re-considered in the light of the current distribution below the • limiting current, as discussed above, and in conjunction with purely convective unsteady-state mass transfer. • Part II. Conditions for validity. 2.4. Definition of the limiting current plateau. • It is useful to distinguish three determinants in the definition of a limiting current plateau: 1. The approach to the limiting current, which would be steep, i.e., complete within approcimately 200 mv,if it were not for kinetic and ohmic • contributions to the overpotential (section 2.3); • 2. the termination of the plateau, by a follow-up reaction (mostly • hydrogen evolution), at a potential determined by the composition of the electrolyte (pH) and the state of the electrode surface; 3.6 the inclination of the plateau,. which is ideally zero but may re*ch considerable proportions due to surface area increase (in metal deposition, in particular of Cu) or, less commonly, due to changes in the bulk concen- tration. •• . . . • 2.. Conditions for valid measurement For a valid limiting current measurement itis obviously desirable that the plateau should be well-defined. From the foregoing itis clear that this can not alwars be achieved; in particular, I. The model reaction may be characterized by slow electrode 'kinetics. • 106 Compare, e.g., figures 3.3 and 4.5 for, resp., copper deposition and ferricyanide reduction.at a rotating disk. The model reaction may change the surface area. This is particularly obvious in copper deposition and leads to poor plateaux at high limiting current levels (Figure 3.6). In Chapter 3 methods will be sought to avoid this effect; but it is not likely that it can be completely • eliminated, at high concentrations of CuSO4 .' The bulk concentration of reactant may be low and will become depleted in lengthy experiments, e.g., in free convection measurements (to be discussed in Chapter 4). On the other hand, certain, conditions can be avoided: Unsuitable placement of the reference electrode, such that a high ohmic contribution is included in the overpotential (Figure 2.4b). Changes in bulk reactant concentration due to variations in gas pressure (02 reduction) or, worse, presence of other species reducible at the limiting current potential (02 in I reduction; 'O• in Cu reduction at low bulk concentration).. . . . .. Unsteady-state mass transfer caused by excessively fast current ramps. or potential scans. This is the subject of Chapter 4. Here it should be noted, however', that in certain conditions where transition-times to the steady-state are long, an actual steady-state may never be reached due to continual change of surface area, or depletion of the bulk concentration. .2.6. Conditions for'valid interpretation. . . . 1. Incorrect values for mass transfer rates are obtained, if migration in the el.ectric field contributes more than a negligible fraction of the 107 total limiting current.. (section 2.2) For several types of diffusion layer and for specific ion combinations this contribution has been calculated and can therefore be subtracted. In the practically important case of acidified cupric sulfate the migration effect is strongly dependent on the dissociation of bisulfate ion. This dissociation appears to be complete, but the experimental evidence is not conclusive 2. Mass transfer rates from limiting current measurements in well- supported solutions should be correlated with ionic, not molecular, diffusivities. These can be calculated from limiting current measurements, e g , at a rotating disk electrode In Chapter 3 this will be done for Cu"' ion in CuSO4 - H 2SO4 - H20. For diffusivities of Fe(CN) and Fe(CN) with various suppoxting electrolytes, see Table 1.4. The diffusivities thus obtained are necessarily effective diffusivi- ties since 1) they reflect a migration contribution which is not always negligible; 2) they contain the óffect of variable properties in the diffusion layer, not accounted for by the well-known solutions to constant- property equations. In the case of Cu ion diffusivities they may also reflect the effect of surface area increase during the approach to limiting current 108 Chapter 3. ROTATING DISK INTEGRAL DIFFUSIVITIES OF .CU 2 IN CuSO -H SO -H 4 2 4 2 0. 3.1. Introduction. In many investigations of mass transfer at electrodes by the limiting current method ,use is made of the copper deposition reaction in acidified CuSO4 solution. To correlate the results in terms of dimensionless numbers. (Re, Sc, Cr, Nu) the diffusivity of the reacting ion has to be known. From an analysis of the electrochemical mass transfer process in well- supported solutions,jt becomes evident that the use of the molecular diffusivity (of CuSO 4 ) is not appropriate. On the other hand the ionic (Cu2 ) diffusivity cannot be determined from molecular diffusivities; it is only known, in the limit of infinite dilution, from the Nernst- Einstein equation. In such solution's, one can calculate, the ionic diffusivity of Cu 2 is approximately 20% lower than the molecular diffusivity of CuSO 4 . Confronted with these unattractive, alternatives, many investigators have used the molecular diffusivities measured by Cole 141 and Gordon for CuSO4 in the ternary system CuSO 4 -H2SO4 -H2O. Their measurents were carried out in diaphragm cells, mostly at 18 °C. These 11 diffusivities have been correlated by Fenech using the Stokes-Einstein equation: . . . -10 pD/T = (2.495 + 0.0173 CHSO + 0.0692 c0) x dyne/ ° K (3.1) 2. This expression, originally given for 22 ° C, has been used extensively in later work. . . . . 109 The validity of (1) for correlations of limiting current measure- ments was first questioned by Arvfa, Bazán and Carrozza,' 42 and later by Wragg and Ross. 63 The latter found that limiting currents in an annular flow cell could be correlated in better agreement with the Lvque mass transfer theory if a lower, mobility (Stokes-Einstein) product was employed: PD/T= 2.09 x 10 dyne/°K (3-2) This value is based on ionic diffusivitjes calculated from limiting current measurements at a rotating disk electrode by Arvfa, Bazn and Carrozza. Effective ionic diffusivities at a rotating disk electrode are calculated from the Levich equation (valid for constant physical properties) by inverting it: • D = v(0.6205 cZF v/i1)3'2 (33) This has been done earlier for fern- and ferrocyanide ions in excess NaOH (or KOH). 47 ' 51 The values of D on which (2) is based, were measured by ArvIa c.s. in four Solutions, of concentrations lower than 0.1 M in CuSO 4 . Practical conditions, e.g., copper refining or winning, often require higher Cu 2 + and H concentrations. The present work is intended to make available diffusi'vity values for a larger range of CuSO4 and H2SO4. 3 2 Rxperimental procedure A. Equipment The disk electrodes used in this work had the dimensions and shape .4 reconanended by Riddif6rd 110 (Figure 3.1).The central part was machined from OFHC copper to a tolerance of less than .001 inch, then embedded in epoxy resin and machined to the specified flare shape. The electrode diameter was 0.3 cm or 0.4 cm, and was measured from an average of 4 readings. by means of a Nikon projection table. . The uncertainty in the electrode area, due to deviations from circularity and to variations depending on the surface preparation, was estimated at + 2%. The disk was attached to a spindle which was fitted tightly in two 7/8 inch O.D. bearings, mounted in a heavy brass case. In the galvanostatic series, of experiments the spindle and disk were driven by a variable speed motor (Bodine Electric Co., Type NSE hR, gear ratio 10 to 1). The motor was connected to an adjustable d.c. voltage supply (Power Designs. Inc, Model 5015A). For reasons of practical convenience rotation rates were limited to 120-400 rpm. The rotation rate was measured by a Strobotac (General Radio Co., Type 631-BL). In the potentiostatic experiments, the spindle and disk were connected by a rubber coupling to the shaft of a Servotek Control Motor with Precision Adjustable-Speed Drive (Series 500). Theexperimeita1 rang.e of rotation rates was 600-1500 rpm. ' The speed settings were calibrated with a Strobotac, accuracy + 1%. The alignment of the disks was ensured by means of a flat annular interface and a screw fitting. The excentricity of the electrode center with respect to the spindle axis was less than 1 micron.' 111 3toinless drill rod toinless sleeve (pressed) Ii gn m ent opper FHC) poxy XBL 715-6774 Fig.. 3.1. Rotating disk and lower end of spindle assembly. 112 The disk was polished by hand before each measurement, either on a glass plate or on a polishing wheel. A slight deviation from perpen- dicularity between disk surface and spindle axis usually developed in the courseof time. When this deviation was larger than approximately 1 ° the limiting current curves would be severely affected, giving irreprodu- .cible and usually too high results. The disk was then machined down and re-cast. The conical shape of the copper rod near the disk surface (Figure 3.1) was found to help maintain perpendicularity since it stiffens the epoxy disk and acts as a heat sink in wheel polishing. As shown in Figure 3.2, the disk was submerged in approximately 1.25 Lof solution contained in a cylindrical cell of lucite. The bottom of the cell accomodated a circular copper anode of 7.5 cm diameter. The reference electrode of copper (OFI-IC) was inserted, in a glass c'ompart- • ment connected by capillary to the main cell. The capillary emerged in the bulk solution 0.5 cm from the outer edge. of the disk, in the plane of the disk. . . The cell was submerged in a constant temperature bath kept at 25.0 + 0 1 ° C Temperature variations in the cell were occasionally larger, due to frequent exchanging of the disk. All recorded currents were corrected for deviations from 25.0 ° C, by a temperature coefficient (di/dfl/i =+0.02 °C. . .. . . . . B. Electrical circuit In the galvanostatic series of measurements the current source was a model 28 Lambda Electronics Corp. regulated power supply. A • Keithley Electrometer (model 610R) was used to measure the current, in conjunction with an x-y recorder (F. L. Moseley Co. model 4B) to plot 114 current vs. overpotential. In the potentiostatic series of measurements • Wenking 61 RH potentiostat was used and the current was recorded as • potential difference over a known resistance R (Figure 3.2), by a Brown-Honeywell Electronik 19 Recorder. Potential control was manual or by means of a digital function generator designed to give a signal of up to 6 v in 4095 steps (or a binary fraction), in a total time variable between 1/128 sec and 12,800 sec. The time-scale of the strip-chart recorder was calibrated so that potentials could be read in slow scans. The electrical connection to the rotating disk electrode was accomplished by means of a mercury well. The resistance between the cathode surface and the terminal of the potentiostat was less than . S. C. Analysis and physical properties of the solutions Solutions were prepared using A R CuSO4 5H20 and H2SO4 CuSO4 concentration was varied from 0.01 to 0.64 M, H2SO4 concentration from 0.3 to 2.5 M. Oxygen was excluded from the solutions by purging with nitrogen. Particularcare was taken with solutions very dilute in CuSO4 . • The CuSO4 concentration was analyzed by electrogravimetry, using platinum gauze electrodes, or by titration with EDTA, using murexide as • an indicator. The acid was titrated with NaOH potentiometrically (end- point pH 4.5 to 5.5). Characteristic aspects of the titration curve - have been discussed eisewhere. 55 The density and kinematic viscosity of all solutions were determined by standard methods, usingbulb pycnometers and Ubbelohde type viscometers. The refractive index for sodium d-light (5893 X) was also measured, by means of an Abbe refractometer. 115 The experimental accuracy of these measurements is estimated to be: density + 0.0002 g/ml kinematic viscosity + 0.002 cst refractive index +0.0001 CuSO4 concentration + 0.001 M H2SO4 concentration + 0.01 M D. Surface treatment Near the limiting current the copper deposit is characterized by locai dendritic growth and creates a very rough surface (see below for some qualitative observations). Therefore, as a rule, the disk was removed after each measurement, or series of measurements, at one rota- tion speed. The disk was then polished with increasingly fine grades of siliconcarbide or emery paper until all traces of the previous deposit, which often persisted as a waviness resembling the texture of orange skin, had 4isappeared. The disk was then further polished on'a glass plate or on a polishing wheel, using lens powder (0.3 ji grain aluminum oxide) or diamond paste (1 .t grain) The maximum scratch on the surface was then 1 i or less. The disk was next thoroughly washed with carbon tetrachloride or ethanol. After rinsing with distilled water and with acidified cupric sulfate solution, the disk was immediately screwed on to the spindle shaft and submerged in the solution The shaft was kept running during this operation to prevent air bubbles from sticking to the disk, although an occasional small bubble at the electrode did not seem to lower the limiting current significantly. U6 The reference electrode and the anode were etched briefly in nitric acid then rinsed with distilled water and acidified cupric. sulfate solution before being inserted. . 118 3.3 Results A. Galvanostatic measurements In a first series of measurements the limiting current was approached galvanostatically. The current was increased in steps, the increments being initially large but decreasing after the overpotential reaches 300 to 400 my. The time allowed after each step for the potential to reach a steady value was approximately 5 seconds or longer. The total time to reach the limiting current was approximately 60 sec. Figure 3.3 shows an example of a galvanostatic curve (dots). The limiting current was taken to be the current value at the inflection point, which occurred between. -500 and -700 my overpotential. In particular for rotation rates above 300 rpm and for solutions more concentrated than 0.1 M CuSO 4 , the limiting current plateau was only distinguishable as an inflection point. The linear dependence of the limiting current on the square root of the rotation rate is shown,.for a number of solutions, in Figure 34 Multiple points at one speed represent measurements in which the electrode was not re-polished. In such cases care was taken not to exceed the limiting current by more than one or two percent. As expected the change in limiting current value compared to the first measurement is cumulative and depends strongly on the level of the limiting current. In concentrated solutions and at high rotation rates the passage to the limiting current was therefore made as quickly as compatible with potential equilibrium after every step. The shortest total time was 12 seconds. A marked 120 improvement in the, quality of the plateau was observed when the passage to the limiting current was shortened The least squares slope through the origin was obtained for all solutiOns using the first, or the first 2, measurements at each speed. The diffusivities calculated according to (3) are collected in Table 3 1, together with the other physical properties As can be inferred from Figure 3.4, the spread of the data for, one particular solution was. rather large and the standard error of the slope 'through the origin is of the order of 2-3.%. The uncertainty in D is then of the order of 5%. . . . . Further, it appeared that even though the passage to the limiting current was fairly rapid in the concentrated solutions, thelimiting current level was still affeëted by the'deposit. Lease-squares slopes, if not forced through the origin, show a negative intercept. As mentioned' above, at high limiting current levels the plateau is only preserved if the time of passage is short. If the "deposit rougnness, though barely , 'visible at 10 x magnification, does affect these galvanostatic measure- ments, a marked dependence on the CuSO 4 bulk concentration 'should exist. Figure 3.5 shows that this is indeed the case'. The mobility produc't iiD/T depends much more strongly on the CuSO 4 concentration than on the H2SO4 concentration. • For comparison some diaphragm cell diffusivi- ties of 'Cole and Gordon are also shown, as well as the values predicted by Fenechs correlation (1). The weak concentration dependence of the latter isin agreement with expectations (see Discussion, section 3.4), but it is clear that, at least for those solutions that are < 0.1 M in 122 '-4 00 V) C) - '.0 00 - LI) co U) C 00 (N C) C) V) C) '-4 -4 0) (71 Il) 00 U) -1 C) N N') '0 (N (N C) 41) '0 '.0 -A N-- C 00 00 C) C 0) 00 00 0) 0) 00 -. e.j CN 1-1 - ,-4 - -4 -4 (N. (N (N (N (D. C) U) c3 0) E C.) E r) 4.4 I3 (N 110 00 N. '0 (N E - '0 -4 'J '.0 Cl 00 0) C (N '0 N- '0 0) 00 (I) -4 N- N- N- '.0 '0 '.0 N') U') -4 00 N- C C) N- - U) '-4 E 14L) C44o (N '0 4)) U) -4 00 - N- -4 (N '.0 N') .-4 00 '.0 00 00. '.0 o 4.) .-4 '0 (M (N '.0 -4 It 00 0) 00 Ln Itt Cd < (N C) E L/) 41) LI) 'cl )") '0 L/). LI) 1!) • + 0 C) (N (N ri 4-O OLI) (N - '.0. N- 00 00 (N 00 (N N C •rI C) (A L/) .4 00 LI) N) '0 N') 0) N- '.0 00 N- 00 LI) 4- 0 .-4 -.. C) C) 0 '-4 (N C) C C) '-4 (N C) C) C) - .4•4(/) • (N . • .. . C - 1-4 '-4 4 C • U U) 44 00 00 (N N- C) - 0 (N N- 00 - 4.4 C ,4 0) 1-4 . 1,- 00 '.0 N- '.0. If) ' N') '0 C)- 00 14 0. E (N '0 (n LI) N') '.0 (3) (N LI) N- -00 C) C) C) 0 - C C C '-4 -4 C) C) 0 - 4)) 0 •-' _4 -4 1-4 _4 _4 -4 _4 . 1.4 ..-I od (/) ,) '0 C) 0) '.0 '' (N .-4 N') '0 0 C N') N') •.-f •r-I C) N- 00 '.0 '' NO C) C) '0 to (N . C) C ti U) (1) X 0) 0') 0) 0) '' 41) C) (N LI') (N . DO C C) -'4 (N C) C) ..-1 -4 (N C -4 - -4 . 'C) '0 N- N') o N'). ) N- C) -4 00 0 0) 0) 0) 0) 0) '.0 '.0 00 '.0 '0 '.Q -4 0) N') • . • Cl) 0 0 0 C C) C) C 0) C) C) 0 0 C) C) C C) •C) • C C) C) - 0 -4. • • • - . . . . • • . - 0 C C 0 C C C C C C) C C C C) N) N') C (N '' C) LI) C '.0 C) . (N • - -4 - N') I.f) - -4 N') ,-4 ' - < • .CC • < , • • .. ... . ...... . .. . . 123 • . .. .. r-. 0 t4) ) • . . . 0 ') 0) - 0) 0) LI) LI) • U) LI) \0 LI) LI) t') N) tn to • . .. . . . • . 0• 0 N) . — -Ict . LI) a •o 0 N) • . . 4 Q 0) 00 LI) \Q '.0 '.0 • • . . . . C (1) N) LI) LI) • ,.- ...... < •0. . 71 E Ln Cd • . C.) .•' ) .. . . 0 U) '1.. 00 00. M. '.0. N) - .• -. 0' 00 LI) . .N) C) LI) • • '.0 . 0 00, • . . . - . • - E LI) LI) LI) ./) ' ' LI) '.0 LI) C) . 0 '.. LI) . N- LI) 0 . '.0. ('4 00 - 0 ., U) '.0 '-I 0) . -4 N 00 0) '.0 -4 0 • - -•--- 4 r'i NI NI N) ('I NI . •...... 0 - - ' 00 00 N. on . LI) 0 '.0 - LI) •. — C N.. U) ('4 N. 00 0 00 N. N) .. . ... . E—.. -NI N) '.0 NI . (7) U) 0) ('I 00 C) 4 . vo-4 . . 00 .. . •. . . . - . . - . - . 4 . -4 . .. C). N) -4 N. LI) C U) cc cc •- 0 r-" ',Q. - LI) LI) (7) CD - N) '.0 00 . C LI) .- 0) 0. U) - . - 0 NI NI •. 0 • -0...... U . . -. U) N) C . U) --4 00 . '0 0 0 NI - . N. ('4 '0 0 0 ('I. 'U) LI) U). .0 If) 1-4 1-4 . -4 • . NI NI '0 •• ..u•...... C 0 0 0 0 0 C C 0 C - N) . . . . .• . . .. .0 ., ••. . - N NI -' 00 0) 0 . • .. ('4 N. ('4 00 • < . < . ... . • 124 Table.3.2.A. Rotating disk integral diffusivities of CuSO4 in H20, at 25 ° C. • ** l/'2 * 10 CuSO4 p DxlO 6 i1./ pDxlO /T (M) (g/ml) (cp) (cm2/sec) (masec 1 " 2/cm2 ) dyne/ ° K 0 0.997 . 0.890 18.45 5.50 0.00998 0.999 0.898 15.19 1.608 . 4.58 H 0.01986 1.0007 0.902 . 14.75 3.135 4.46 0.04969. 1.0055 0.920 13.68 7.443 4.22 0.1000 1.0135 0955 12.63 14.13 . 4.04 0.2000 1.0290 1.02 11.72 26.66 4.01 0.2985 1.0444 1.09 11.23 38.34 4.11 0.4000 1.0602 1.16 11.06 50.47 4.30 0.4989 • 1.0756 1.24 10.67 61.07 4.44 • from Ref. 96 (Table 2.3) • • •**• • • • • • • from Ref. .139 (Figure 2.3) • *** • from Ref. 140 • B. Diaphragm cell diffusivities of CuSO4 in CuSO4 -H2SO4 -H20, at • • • 18 °C. (after Cole and Gordon 141 ). . p* CuSO4 • H2SO4 • Dx106 • i.iDxlO1 9/T • • (M) •• • (M) (cp) • (cm 2/sec) dyne/ ° K 0.05 . 1.5 1.44 5.21 . 2.58 0.10 . 1.48 • 5.15 2.63 0.25 • . . 1.61 5.02 2.78 * from Ref. 97. . ** . from Fig.. 8 of Ref. 141. . . . . 125 CuSO4 , the predicted values are too high by as much as 20% This is in agreement with the conclusions of Wragg and Ross. 63 The values of the binary CuSO 4 diffusivities are also shown, but as integral diffusivities according to (3) where i lim includes the migration contribution (Table 3.2). The migration contribution to the limiting current is at least partly responsible for the dependence of pD/T on the CuSO 4 concentration. This will be considered below in more detail. To investigate whether the deposit effect could be eliminated more completely, a second series of measurements were made, using potentiostatic current control. • . An interesting observation made during the galvanostatic measure- ments.was that the building up of the deposit does not always lead. to an increased limiting currents. When measurements at one particular speed were repeated without.re-polishing the disk,the limiting current in concentrated solutions (> 0.25 M) would usually increase, in solutions of> 0.05M CuSO4 a slight decrease (approximately 1%) was observed in the next two or three measurements. Eventually the electrode surface becomes very rough and the current increases again. B. Potentiostatic measurements potential scan In these measurements the limiting current was at first approached by potential scanning The potential scan rate was -32 or -40 mv/sec The limiting current was thus reached in 20 to 25 seconds This rate was chosen after preliminary experiments with various rates, in order to minimize non-stationary mass transfer effects (see Chapter 4) Typical l].mlting current curves are shown in Figure 3.6. The rotation rates were varied over a larger range, from 60 to 1200 rpm The solutions employed 126 li > Ic 0 Cr 6 2 0 200 400 600 800 1000 = 7/(mv). TYPICAL LIMITING CURRENT CURVES XBL7I3- 6632 Fig. 3.6. Typical potentiostatic limiting current curves. (R 0= resistance of nieasuring resistor). 50 40 E 0 20 I0 0 128 were not identical with those of part A. They were generally of a higher CuSO4 concentration, so that the deposit and migration effect would be more prominent. The potentiostatic curves show a plateau only at low current levels At higher current levels the plateau appears to become shorter and increasingly tilted, but' the potential at whichhydrogen ion reduction starts to increase the current, is still recognizable. The potential of this inflection point depends of course on the acidity of the solution, but.jt also •appeared to depend on the rotation speed and on théCuSO 4 concentration of the solution, i.e., on the amount of deposit (Table 3.3). This behavior had already been found in the galvanostazic. measurements, but there the inflection potentials were, in general, less negative than in the •present measurements. . It was also clear from visual inspection that at the potentiostatic inflection point the copper deposit had become quite rough (dendritic), while in the galvanostatic measurements the surface is still relatively smooth if the current passage is terminated at the point of inflection. Therefore, although the currents at the inflection point were surprisingly linear with the square root of the speed (Figure 3.7), they were not limiting currents corresponding to the smooth electrode area. This was also obvious from a comparison with galvanostatic limiting currents (Table 3.3). The difference in current level at the inflection • point appeared to be related to the amount by which the potential was more negative atthe potentiostatic inflection points. 129 Tab 11p 3.3. Potentials. and currents at the inflection point in acidified 0O5 M CuSO4 . . . . . inflection point potential (mv) . Ratio.of (A.) potential scan (B) galvanostatic currents rpm 200 600 1200 260 A/B M.H2O .. .. .. . . ., .. . . . . .... .. 0 5 760 780 810 660 1.20 1.0 700 750775 . 660 1.16 1.5 650 675 750 650 1.03 20 . 600 650 725 600 1.06 2 .5 . . . 580 .. . 130 C. Potentiostatic measurements: potential step To avoid the effect of deposit roughness on the limiting current, a potential step to the limiting current would be most effective. Two. difficulties are associated with this.procedure: 1. the potential necessary to reach the limiting current depends on the rotation speed; 2. a certain transition time has to elapse to avoid unsteady-state mass, transfer. These points will be considered in more detaiL Ad 1. Potential range of the limiting current. From galvanostatic as well as potential-scan curves it is evident that the limiting current plateau shifts to higher potentials as the limiting current level becomes higher.. This is an indication that the reaction kinetics of the copper deposition reaction contribute an appreciable surface oerpotential to the total overpotential. The approach to the limiting current plateau is thereby made more gradual. Under certain simplifying assumptions the steady-state limiting current curve can be constructed (see Chapter 2 2), and in particular it can be shown that for well-conducting solutions the approach to the plateau depends on two parameters: the relative magnitude of limiting current and exchange current density (M = i 1 ./i), and the cathodic transfer coefficient. (Figure 2.6) . . . . The potential step to be applied wastherefore based on the expecta- tions according to this model. The exchange current density was estimated from some of the galvanostatic curves Figure 3.3), by means of the potential at half the limiting current (equation 242). ' If the 'value of ' was assumed to be 0.25, the average i 0 was approximately. 2 ma/cm 2 for a solution of 0.1 M CuSO4 . This is somewhat below the values found in. ' ' . 131 electrode-kinetic, studies using extremely pure solutions. 134,138 Since the parameter M varied between 10 and 500 for the range of concentrations (0 05-0 5 M CuSO4 ) and rotation rates (100-1200 rpm) in this work, the potential step was set at -500 my This is large enough to ensure a near-limiting current even in the most concentrated solutions, and is below the minimum potential for hydrogen evolution even at the lowest current levels. . Ad 2. Mass-transfer transition time. A large enough potential step applied to a cathode would cause the initial current to be infinite if the reactant concentration at the electrode were to become instantaneously zero at the electrode Actually the current excursion is limited because the measured over- potential contains an ehmic contribution and a surface overpotential. Both of these retard the initial concentration drop at the electrode On the other hand, the time necessary to reach steady-state conditions is determined primarily by the diffusion mechanism It may, however, be dependent on the level of the initial current, i e , on reaction kinetics' and ohmic resistance. At a rotating disk electrode convection reinforces the diffusion process A complete solution of the current response to a potential step at the surface would therefore be quite complicated Since we are here only interested in the duration of the non- stationary process, let us assume that it is determined only by the convective-diffusion process This is the case if the time-period during which the reactant concentration at the electrode has not yet reached a virtual zero value, is very short compared to the total transition time 132 For a large potential step (to 0 = - ZF/RT) ata cathode in a stagnant medium, Gerischer and Vielstich 143 give the characteristic time-period 222 DcnF 2 De = 2 -2 2 (3.4) k 0 At times >> t the current is the same as if the concentration at the electrode had been zero from the beginning. For the copper deposition reaction and a potential step to -500 my., t 1 has a value of the order of 10 sec On the other hand, it can be expected that at very short timesthe concentration transients at a rotating disk electrode approach those in a stagnant solution,since the concentration change takes place so close to the surface. A numerical solution for the case of a high Schmidt number shows that, up to a characteristic time-period t 2 = 0.16 Sc 1'3 C , ( 3.5) the flux response to a concentration step at the.disk surfáceis, within 1%, identical with that in a stagnant medium. (See Appendix B and Figure. B.1) The period t 2 , for solutions of Sc 2000 andrótation rates between 200 and 1200 rpm, is of the order of 10_i .sec or higher. We can therefore safely use the convective-diffusion transition times to a flux within 1% of the steady-state value. According to the numerical solution given in Appendix B, this transition time is: 133 Table.3.4. Transition times for mass-transfer at a rotating disk (In sec). .7 . 2 • - • • • ui • • • conc. • step • current step. (rpm) • • • • Sc . • Sc Sc Sc 2000 • •. .5000 • 2000 • 5000 50 4 94 8.45 13.16 22.50 • • • • 100 • • 2.47 •. 4.23 •. 6.58 • • 11.25 • • • • 400 : • •. • 0.62 • . • - 1.05 • •• 1.64 • 2.1 • • • • 1600 • • • • 0.15 • 0.26 0.41 • 0.70 1 T = wt/Sc 1'3 = 2.05 at i/i ss = 1.01 (eq 6) • 2 T.= wt/Sc 1 • . • ,• := 5446 at (c e 1_cb )/ (c e l ss c b.) = 0.99 (Ref.. 154) 1,-i I I I V 1200rpm 0906 o5 404 Cr 0112 CuSO4 = 0.20M A0.1285cm2, R0=0.419.n. (°) 0 I I I C I• 2 3 4 H t(sec) 0.2 CP • log (wt Sc') XBL7I5-6782 Fig. 3.8. Transition followinga potential step to-500 my, for various rotation rates as indicated (part a). In part (b) the various • curve are combined into one, using iuf' 2, and a dimensionless time T. R0=value (Q) of precision resistance (Fig. 3.2) vertical marks. in (a) correspond to T 3 (3.6). 135 t = 2 05 Sc 1/3 W 1 (3.6) ThéSc number necessarily has to be estimated a priori, but t is not very sensitive to its value. Using a potential step to -500 my and observing a transition time acccrding to (6), the current was now measured again for all solutions, at various rotation rates. Before the potential step was applied, the disk electrode was plated at 200 rpm and at a constant current of approximately 10%, or less, of the limiting current (50 ma-mm/cm2 , i.e.., approximately 1 p thickness). . . . . . The currents were satisfactorily linear with the square root Of rotation speed, except above 600 rpm where they appeared to be low (Figure 3.7). From Figure 3.8 it can be seen that this is due to 1 The limited response-time of the recorder; 2. the fac,t that the current does not reach a steady value after the transition time (6), but keeps decreasing for several more seconds (apparently depending on the current level). For dilute solutions a fairly, steady value establishes itself, while in concentrated solutions the current starts to increase again, apparently due to roughness Ibl and Schadegg 150 have observed a similar behavior even at 2000 rpm the current decrease lasts for approximately 30 sec From impedance measurements they concluded that at currents close to, but below, the limiting current a surface smoothing or re-crystallization takes place during several minutes. The consequences of this extended transition-time will be discussed below. For the calcu- lation of diffusivities it is assumed that the currents measured after 136 1-4 *1) (N ) t (N - LI) LI) 0) 0) O N- C) 0) 0) 0 U) m 00 t10 LI) 0 0) LI) 4) C) LI) .LJ) LI) U) LI) LI) I) N) ) I) tn t1) II) tI)M t) 41 . .. . o - 4 4 4 4 ,4 0 ci • 0 0) 0 N. 00 vo 0 N '.0 Il) 00 -4 IRt oo - 00 U) r-4 0 1.1) 4 C) ) -4 0 IJ LI) - '.0 00 ' \P 0) C) -. r- 0. 0) C) - ') '0 r'i N C) LI) 0 0 0) t') PI) (N tI) 0 (I) (N E E U 00 LI) '0 (N (N (N N- 0) '.0 U) 0) .-4 00 - (N LI) '.0 C) LI) P') '0 N- (N LI) '.0 '0 - N '0 • U I) P') 0) 00 0) '0 00 0 '.0 '.0 J LI) '.0 U) - - -1 (N P') (N (N (N P') o ..- (I) I) 0 E o U - '0 ()) ..-i N- V, .-4 00 N- qlT N- P') 14 C) '.0 (N i-i U) 0 11) N- LI) N- '0 0) 0 PI) '0 P') 0 1-4 '0 C) F- (N (N '.0 LI) ' '.0 '.0 N. L/) N 00 '0 '0 '.0 '.0 F-- C) E o U -.4 + 138 the transition time (6) are actual limiting current.s•corresponding to the measured surface area. Table 3.5 gives the.integral diffusivities from these potential- step measurements. In Figure 3.9 the mobility product is plotted against the CuSD4 concentration. The dependence on this concentration is comparable to that in the galvanostatic measurements, although the values themselves are generally higher. Also, the spread in the data is larger. 3.4. Discussion: Migration Effect The diffusivities calculated from the galvanostatic and pótentio- stat:ic measurements can be corrected for the effect of migration if some assumptions are made concerning the degree of dissociation of HSO ions. The few experimental data available concerning the composition of the electrolyte at the cathode, in free convection and in pure dif- fusion, suggest that the HS0 ion is almost completely dissociated. Since the dissociation constant of HSO ion in very dilute solutions is small (K8 0.0104 at 25 °C), the apparently almost complete dissociation in concentrated solutions is due to the low activity coefficient of and SO4 ions: . . ' KI = K8f / £ - (3.7) HSO H SO4 where Kj is the stoichjometrjc dissociation t'constant" and KB the therjno- dynamic (true) constant. From spectroscopic data on I-1 2SO4 solutions it is known that KBI varies by 3 orders of magnitude as the concentration increases from 0 to 3 M. . 139 0 0.5 1.0 15 c"2 (M "2 ) H2 SO 4 XBL7I5-6784 Fig. 3.10. Degree of dissociation of HSO ion (expressed as I/Ku) as a function of H2SO4 concentration, for various CuSO4 concentrations. Complete dissociation of CuSO4 is assumed, except for 0.30 M (dashed line) where the effect of partial association is also shown 140 Since the presence of bisulfate ions, even in small amount, increases 2+ the migration current of Cu ions considerably, it is important to establish the degree of dissociation of the bisulfate ions more accurately. Hsueh and Newman have correlated the results of spectroscopic investiga- tions of dissociation in aqueous H2SO4 solutions as follows: 5.29/I = r P n(K'/0.0104)B (3.8) 1+0.56,'I r where: 'r is the true ionic strength (in mole/2). Assuming that this relationship holds also in solutions containing CuSO4 in addition to H2SO4 , the degree of dissociation can be calculated and expressed in terms of the parameter I/K, which is dimensionless (I is the stoichiometric ionic strength). The method of calculation is given in AppendixC. Figure 3.10 presents I/K as a functionof CuSO 4 and H2SO4 concentration. The value of I/Kj is close to 5 for the range. of compositions (0-0.5 M CuSO4 , 0.5 - 3 M H2SO4 ).used in limiting current experiments. . The contribution which migration makes to the limitIng current at a rotating disk has been computed by Newman and Hsueh for I/K values of 0 (complete dissociation of HS0), 10,100 and (F1S0 undissociated). Additional computations have been made for I/K = 5; for the details see Appendix D. Figure 3.11 shows the ratio of limiting current to limiting diffusion current vs. the ratio of H2504 to total electrolyte coñcentra- tion. The migration contribution is approximately two times that in the case of complete dissociation of US0 (Table3.6). . 142 Table 3.6. Corrected diffusivities and true ionic strength (present work • and literature data). . . . . - CuSO4 H2SO4 I pDxlO 10/T,corr. (M) (M) (M) (dyne! ° K) A9 00093 0.476 0.764 1.825 A2 0.00956 0.980 1.573 1.795 • A4 0.00976 1.469 / 2.388 2.067 All 0.00947 1.946 3.196 2.025 A13 0.00943 2.434 4.032 1.957 AN 0.0460 0.492 0.925 1.719 A3 . 0.0461 0.991 • 1.727 1.768 AS • .0.0488 1.463 2.526 • 1.948 Al2 0.0462 1.936 3..327 1.921 A14 . 0.0466 2.420 4.158 1.845 A39 • 0.0967 0.500 • 1.125 2.029 AlS 0.1017 1.003 1.959 1.987 A40 0.0997 1.253 1.652 2.138 A16 • • 0.1031 1.514 .2.820 2.082 A41 0.1001 1.769 3.245 2.124 A17 0.1024 2.013 3.671 • 2.155 • A42 0.1015 • 2.551 4.600 2.234 • • A6 • • 0.2273 1.457 3.203 • • 2.172 A18 0.2520 0.995 2.536 • 2.131 • •. . A19 • 0.2565 2.006 4.260 2.376 • . . A20 . 0.2561 . 2.535 5.200 2.355 A7 • 0.4018 1.464 3.926 . 2.374 A21 0.4536 0.988 3.341 2.204 143 Table 3.6. (continued) : CuSO4 H2SO4 pDxlO'0/T,corr Tr (M) (M) (M) (dyne! ° K) A8 0 6394 1.468. 4 883 2.613 Bli 0 0500 0.492 0.941 2 033 • •. B18 . . .. 0.0498 0.987. . 1.736: • . 1.976 • • • . B13 • 0.0499 • • . :1.493 •. 2.582 • 1.866 • B12• 0.0502 2.013 . 3.468 2.005 • • B8 . • • • . • • 0.0992 0.496 • 1.129 • • • •. 2.150 • B5 0 1017 1 008 1 967 2.160 B68 0.2043 0.498 1 545 2.156. B4 • • 0.2005 . • • 0.992 • .. 2.326 • 2.007 • • . • B1O. 0.2956. ..... 0.295 . . • 2.07 . •.•. 1.864: 97 • : •: 0.2998 0.494 . 1.912 . 2.2.91 B2A 0 3158 1.011 , 2.810 2 184 B14 0.295. 1 1 492 3.539 2.716 : B1$ • • . 0.299 . 2.020 4.456 .. •; . 2.786 . • I B16 •. . • 0.4002 0.991 . • 3.117 2.411 B9 0,8001 0.9968 3.525 • 2.671 . . • . . • . . . •. . • . • . . • • . . 144 Table 3 6 (continued) .142 Arvia, Bazan, and Carrozza T CuSO4 H2SO4 I UDx10 10/T UDx10 0/Tcorr ( °C) (M) . . (M) (M) (dyne/ ° K) (dyne/ ° K) 24.6 0.011. 0.715 1.15 2 026 2.005 18.0 0.025 1.693 2.82 2.094 2.072 25.0 . . U It .2.070 2.048 24.9 0.048 1.382 2.41 2.200 2.150 30.4 " . . 2.026 1.980 35.0 II I! . U 2.118 2.070 41.8 U " 2.214 2.163 18.0 0.077 1.752 3,11 . 2.011 1.955 24.7 2.067 2.010 Capillary cell diffusivities (Hsueh 96) CuSO4 H2SO4 1r liDxlO 10/T JiDxl0 10/T,corr (M) (dyne/ ° K) (dyne/ ° K) 0 1024 0.810 1.62 1.970 1.793 0.1008 0.888 . 1.76 1.97.0 1.811 0.0986 0.998 1.95 2.00 1.856 • 0.1005 1.085 2.09 2.02 1.882 0.1006 1.195 2.27 2.02 1.8.97 0.1006 . .1.295 . 2.44 2,03 . 1.913 . . . 0.0999 1.375 2.57 . . 2.05 1.940 0.0999 1.473 2.74 2.06 1.957 only the data for (H2SO4 ) > 0.5 M are included 145 The integral diffusivities, thus corrected for the migration con- tribution: 3/2 DID = (3.9) are shown in Figure 3.12 versus the true ionic strength (calculated as in Appendix Q. Table 3.6 contains all the data shown in Figure 3.12, including.the data of Arvia a.o.., and the capillary cell diffusivities measured by Hsueh. . The latter have been corrected for the migration effect in stagnant diffusion, as computed by Hsueh and Newman. In spite of the migration correction a marked dependence on the ionic strength persists This dependence is only p8rtly due to ionic concentration effects... FromTable 3.6 the dependence due to concentra- tion effects can be estimated by considering sets of results for fixed CuSO4 concentrations; on the average: . . d(uDX1O 10/T)/dI = 0.05 (3.10) whereas for the entire set of rotating disk diffusivities: (iDxl0 10/T) = 1 792 + 0.119 'r orr (3 11) This outspoken dependence on the CuSO 4 concentration is also evident in Figure 3 12, where the open symbols represent CuSO 4 concentrations > 0 1 M The average of the data given solid circles and triangles is (pD/T) Corr = (2.01 + 0.13) x 10 ° (dyne/ ° K) , (3.12) I- I •* U 0 0 V 0 * AA9 • • o A.1 .. 0. LclLLARy CELL 0 I I I I I • I .0 ..I • 2 3 4 . 5 TRUE IONIC STRENGTH (M) XBL7I3-6630 Fig. 3.12. Integral diffusivities of Cu,. corrected for ndgration effect, as a function of true ionic strength,. 1r Circles and triangles indicate valu 2reported in this work, squares indicate results of ArvI 1 c s Diaphragm cell diffusivities according to 96 Fenech, and capillary cell diffusivities according to Hsueh are also shown. . 147 i.e., very close to the average of Arvia's data (solid squares): 0 2.05 x 10 dyne/ ° K. The average of data for 0.1 M .CuSO4 : 2.11 x 10 • dyne/ ° K is significantly higher than that of the capillary cell data for the same concentration, 1.88 x 10T 10dyne/°K. The average of data for • > 0.1 M CuSO4 is 2.33 x 10 0dyné/ ° K,and significantly highr than that of the data for CuSO4 concentrations < 0.1 M. Note that these values for the averages, and for the concentration dependence, refer to the corrected diffusivities and that the migratiOn correction is by no means unambiguously established. In the concluding section average values and other expressions for the diffusivity will be given which are practically more useful. Although the possibility is not exclu4ed that the migration current is actually larger than predicted (see Appendix D), it is likely that the particular dependence on CuSO 4 concentration reflects an increase in area due to deposit roughness. Thus, in spite of the precautions taken in the potentiostatj.c experiments, the influence of depOsit roughness • seems to be present. The following • section will be devoted to • . this aspect of the measurements. 3.5. Discussion: de,osit roughness • • Theeffect of.deposit roughness has been found, in the course of • • this work, to be quite important except in solutions with low CuSO 4 • • content (< 0.1 M). Several observations have been made: • 1. In potentiostatic scanning and in galvanostatic stepped current increase the limiting currents are apparently high due to an increase in effective .area; however, much more so in potentiostatic scanning than in the galvanostatic stepped increase. • • • If, the mobility, product iiD/T is in reality independent of the ionic strength, then the increase in effective area is from 0 to 15% in the galvanostatic method., and up to 30% in the potentiostatic scan method. The potential step method gives values that are comparable. to the galvanostatic stepped increase method, in spite of the much shorter time for roughness to form 4.. The higher the values observed, .the,more spread (irreproducibility) in the data. . . S. In the galvanostatic technique, repeated measurements with the same disk may initially lead.tolower values. . . Of these observations (1) and (4) are readily explained In potentiostatic scanning approximately 1/2 of the total scan time is passed in the neighborhood of the limiting current (see figure 3 6), while in the current stepping experiments this time was usually 1/3 or less Galvanostatic scans could possibly reduce it even more Ibl and Schadegg, 0 and earlier Emelyanenko and Baibarova, 152 have observed similar differences between potentiostatic and galvanostatic limiting currents. The poor reproducibility of the potentiostatic limiting currents is due to the rough deposit, and is reinforced by the current fluctuations characteristic for potentiostatic control in the limiting current range. The reproducibility is best onpredeposited electrodes and worst on those that are merely polished and cleaned. , Observation (3) is least expected and suggests that a rough deposit is formed in the initial unsteady-state period, followed by re-arrangement 'whereby the surface becomès'more smooth, as mentioned' before (seeFigure 149 3.8a). Figure 3.8b shows the transients for various speeds compared in * one graph with the theoretical transient current (Appendix B). This, too, suggests that the steady-state limiting current (dashed line) as well as the unsteady-state current, is too high; rearrangement appears to take place faster, the higher the deposition rate • It would then appear to be preferable to take the second, surface- rearrangement, transition time as the period after which a valid limiting current value is recorded, instead of the time t 3 (equation 6). However, there is no assurance that the surface is really, smooth; at most a certain minimum effective area may be reached before new roughness formation (of a dendritic kind) starts. Two experiments made in the course of the potentiostatic experi- ments are of interest since they tend to confirm the fairly high effective- area estimates of observation (2). 1. In several measurements the potentiostatic step to -500 mv was followed up:with a slow scan at the rate of -20 mv/sec, through the limit- ing current range. It can be assumed that the current increase measured during this scan reflects the increase in effective surface. The current increase was of the order of 10 to 15% for a 10 second period, in 0.05 M Cu804 solutions (compare Table 3.3). In these experiments it was found that, except for the most concentrated solutions (in CuSO 4 ), the current increase was proportional to the rotation speed Figure 3.13 shows this for the case of solutions Bll (0.05 M CuSO4 ) and B14 (0.3 M CuSO 4 ). This result implies that the increase in effective surface is proportional to the square of the "smooth" limiting current Such a relationship can be made plausible from a microscopic model of preferential growth. • 0 2 4 6 8 10 2 14. 16 18 Vw (sec") XBL 715-6786 Fig 3 13 Effect of rotation rate on the increase in current, between -500 my potential and the inflection point, for a slow potential scan (20 mv/sec) during the time interval At indicatéd = current at -500 my potential, 'e = current after time At 151 In a very simplified mo4el of roughness formation, spheres of diameter a grow preferentially on a fixed number of sites, n. Deposition on the smooth surface goes on side by side with that on the protuberant spheres, but.the spheres present a growing surface. Assume that each sphere contributes an extra area 7ia 2 to the mass traiisfer surface. This implies that the diffusion layer is not disturbed, it is equivalent to assuming a local current increase by a factor of approximately 5. The fraction covered by dendrites, 0d is then related to the effective current Iby = 0.25(1 eo"o = 0.25 M/A , (3.14) where Iis the "smooth" current, A the smooth area, and AA the increase 0 0 in area If a ring adjacent to the edge of the disk would be occupied, then A/A = 0 20 would correspond to a "solid" ring with inner radius 0.975 times the disk radius The growth equation of the protuberant spheres is now 3 (63.54)1(0 A d (n1Ta do M) t\ 6 1 = (2)(8 96)F Since OdAO = nrra2/4 and A = nira 2 , this yields a = 9 2 x 10 8 i dt (cm) (3.16) where i expressed in ma/cm,2 and t in sec If the current is a constant 152 ',IOOO my 400 RPM : 120- 25°C /900mv - 16 mv/sec / 800mv 100 Deposition 700mv - Il/I no 80- ideP0t c'J I 0 - • LIII, o 40- / Hydrogen Evolution 20------ I 0 200 400 600 800 1000 •17(mv) XBL-7Ii-03 Fig 3.14. Hydrogen evolution on a rotating disk previously plated by • negative potential scanning to the terminal potentials • indicated. • CuSO concentration of plating solution 0.3 M; H2SO4 concentration 1 M (during plating as well as hydrogen • evolution). 153 limiting current density, one has after 10 sec, for solution B14 0.84 (at 62rpm) or 3.0 p(at 1500 rpm). The latter dimension is still less than 1/4 of the diffusion-layer thickness at that speed (13 vt). Equation 16 shows that the effective increase of surface is propor- tional to the square of the limiting current, as long..as nodisturbance of the diffusion layer takes place. This condition obviously limits the validity of the model, in particular at high deposition rates. There are also other objections, in particular that it does not account. for non-uniform dendritic growth, which is always observed starting at the edge. This is probably due to non-uniform current distribution on the disk, not to nonuniform availability of nuclei for dendritic growth. The density of nuclei calculated from the slope of Figure 3.13 (sol. B14) is of the order of 0.1.- 1 per square micron. Nevertheless, the model shows in semi-quantitative terms how the preferential sites may lead to a very large increase in effective area, without any hydrodynarnic effects such as local turbulence (which becomes effective only at local Re numbers above 10, i.e., several orders of magnitude larger than the dimensions calculated above). 2 To determine the influence of the deposit on the hydrogen evolution reaction, a potential scan was made at a copper disk electrode rotating in an aqueous solution of H 2SO4 of the same con cen tration as in • the acidified cupric sulfate solution in which the disk electrode had previously been plated The disk electrode had previously been plated by potential scanning (at the same rate) down to various terminal potentials (see Figure 3.14).. The disk was always kept in contact with solution 154 Scans were also made with the disk freshly pre-deposited (at 10% of the limiting current), and after mere polishing andcleaning. Figure 3.14 shows that the rough surface created above 600 my leads to an earlier hydrogen breakthrough. The most straightforward explanation of this phenomenon is the increase in effective surface area. If the 2+ small residual limiting current of Cu i ons (due to spilling during transfer, and to diffusion from the reference electrode compartment) is subtracted, the hydrogen evolution currents yield Tafe1plots (Figure 3.15) with an average slope d2n. = 0.585 If d (3. 17) • somewhat higher than the value = 0.5 generally accepted for hydrogen evolution. Using this 8 value for all recorded curves, an apparent exchange current density can be found as given in Table 3.7. The values of i 0 suggest that already at 700 iw the increase in surface area due to current passage (from 500 - 700 my in 12.5 sec) is more than 30%. It is not surprising that this is higher than suggested by the current .increase in the potentiostatic copper deposition. The current distribution.for the hydrogen evolution reaction can be •expected to be much more non-uniform than for the copper-deposition reaction. The hydrogen reaction is very far below its (hypothetical) limiting current but into the Tafel region, so that the appropriate current distribution * To convert to hydrogen overpotential, use is made of: • E(Cu/Cu) =0.289 V for 0.3 M CuSO4 (stoichiometric activity-coefficient 0.08) - '7 nv vs. H2 /2H IM) 40C) 500 600 700 IMH2 $04 25°C 400rpm Disk 4mm dia. 0 N • E C) • 0 E o Smooth disk Disk plated to: 0 -700 mv(Cu/Cu) - 900 - 1000 0.1 157 parameter is ir ZF 15 RTn Z (318) and a fairly non-uniform current-distribution exists As a result, the region near the edge of the disk where the dendrites predominate, has a higher weight in determining the effective area for hydrogen evolution. The lowering of the breakthrough potential of.hydrogen evolution suggests an explanation for the slightly lower limiting currents observed whengalvanostatic measurements were repeated with the same disk. As the current is forced up, at fairly low potentials an appreciable fraction of the current will be due to the hydrogen reaction; due to the larger effective surface the original limiting current will not be a limiting current any mOre, and this, too, will depress the potential. As a result, the inflection point which was originally the end of the plateau, shifts to the onset of the plateau and, thereby, to slightly lower current • levels. 3.5. Conclusions 1. In solutions which are sufficiently dilute in.CuSO 4 so that migration and surface roughness• effects can be ignored,the rotating disk ionic diffusivity is not significantlydependent on the ionic strength and can be represented by 10 D/T = (2.01 + 0.43) x 10_ dyne/ ° K (3.20) This is based on 23 diffusivity data for the condition c • CuSO4 < 0 1 M and CH so > 10 CCSO . Included are 14 data obtained by galvanostatic 2.4 4 158 • and potentiostatic.techniques in the present study, as well as. 9 data • points of Arva c.s. 2 It should be noted that the latter authors 142 found that addition of glycerol apparently increases the ]JD/T. product of solutions dilute in CuS0. (See Figure 3.17) 2. In limiting current measurements using solutions concentrated in CuSO4 ,the galvanostatic approach to the limiting current is preferable; potentiostatic techniques lead to higher levels of surface roughness. 3 In solutions of concentration > 0,1 M CuSO 4 diffusivities obtained by the galvanostatic approach are not significantly dependent on the H2SO4 concentration, and can be expressed with sufficient accuracy by: (2.03 + 1.82 c 0 ) x 10-10 dyne/ ° K , (3.21) - 4 This expression, based on 14 galvanostatic data of this work is believed to represent the integral diffusivity in more concentratedsolutions with an accuracy of + 10%, taking into account migration and minimal surface roughness effect. 4. The diffusivities found for dilute solutions of CuS0 4 , of up • • to 0.1 M concentration, lie appreciably below the diaphragm-cell .diffusivities of Cole and Gordon., and the correlation based thereon by Fenech. In the range of concentrations 0.2 to 0.5 M CÜSO 4 , frequently used in free convection studies, the agreement is somewhat better., especially if migration is corrected for. This may explain why these • .. diaphragm-cell diffusivities could be used sucàessfully in certain • 159 free-convective mass-transfer• studies. * 5. The rotating disk , ionic diffus.ivities reported in this work appear to be somewhat higher than the stagnant diffusion(capillary) ionic • diffusivitjes reported by Hsueh, 96 for the same CuSO4 concentration (0.1 M). Considering capillary diffusivities in solutions with CH SO > 0.5 M, 24 and correcting for migration, the average rotating disk diffusivity is approximately 12% higher than the average of the capillary diffusivities, i.e., significantly different at the 99% probability level. * • It should be noted that the values reported by Cole and Gordon for the diffusivity of CuSO4 in absence of H2SO4 , at 18 °C, are suspiciously high. With a temperature coefficient D dD/dT = 0.03, they would translate to values at 25° C which are 20% higher than those measured in recent work (see e.g., ref. 139). 2 2. -o 0 0 . 2. 2.( 161 Chapter 4 UNSTEADY-STATE EFFECTS IN LIMITING CURRENT MEASUREMENT Part I. Reviw 4.1. Introduction lla,156 Previous investigations carried out in this laboratory on mass transfer rates along planar electrodes, revealed that the length of time in which the limiting current was attained influenced the limiting- current curve. This unsteady-state effect was noticeable both in the definition (or lack of definition) of the plateau, and in the current value read from it (or from the inflection point) The effect has been most prominent in free convection but, as will be shown in the section 3 , it can be quite important in forced convection also. A comparison of theoretical informatiOn aboUt transients inconvec- tive diffusion with selected experimental data available in the literature, shows that convective transients, in particular on elongated electrodes, are coaplicated considerably by the presence of an electric field. Besides this interaction there is •often the possibility of an unsteady statedue to surface area or bulk concentration changes. For a situation where these complicating effects are not important, namely for a redoxreaction ata rotating disk electrode, some experiments are reported in Part II. The purpose of these experiments is to establish the minimum time required to reach a steady-state limiting current in a representative case of forced convection. 162 • 4.2. Free convection Figure 4.1 shows limiting current curves recorded by Hickman 156 • at a horizontal cathode facing upward, in free convection. The apparent limiting current value is definitely dependent on the time passed to reach the limiting current, it increases 80% as the current increase rate is varied from 0.25 to 20 nla/cm 2min. Figure 4.2 shows that the relative increase in the apparent limiting current is even more important in pure diffusion, i.e., where the cathode plate faces downward. In this case the expectation is that the apparent limiting current will drop to lower and lower values as the current application rate, di/dt,is made smaller. If it were not for edge effects (advection) at the embedded plate, no steady limiting current could be expected In the case of free convection, the apparent limiting current will not drop below the steady-state value as the rate, di/dt, is decreased Hickman, in experiments on free convection at horizontal electrodes, found that a time • ss = >8mm • • (4.1) would be sufficient to guarantee steady-state limiting current values Earlier, FenechU had produced limiting currents in the same free- convection system by means of linear increase of the applied potential (cell voltage) Using concentrated solutions, he found that the limiting current value is essentially independent of the initial voltage applied, 163 I . .1. = 20rna/cm 2 min ôt 0.235 4 0.41 10 5 0.64 3 1.12 E U 2.5 2.5 0 E 6.9 Timeto reach 2 i 8.8 mm - 0.5 0.375 0.25 0.0I7MCuSO4 I. 225 M H2SO4 1.26 Mglycerol 10 x 10 cm cathode 1 cm separation Free convection 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 LE (volts) XBL 698-1210 Fig. 4.1. Limiting current curves recorded for various current application rates in free convection at a horizontal electrode (from Ref. 156). 7 6 CN E LI 5 0 E. 4 3 2 0 0 Oi 0.2 0.3 0.4 0.5 0.6 0:7 0.8 Cathodic overpotential (volts) MUB-12040,A Fig. 4.2. Limiting current curves recorded fOr various current applica- tion rates in pure diffusion at a horizontal electrode (from Ref. 156). 165 provided the same rate of increase is used Starting with tE = 200 my, steady reproducible limiting currents were obtained if t= 500 mv/ > 3 nun (4.2) This time is considerably shorter than the galvanostatic minimum time (1). Since a theoretical analysis of this type of unsteady-state free convection offers severe difficulties, there is no way to confirm this difference except by controlled experiments of the kind undertaken, on a limited scale, by Hickman and Fenech Even in the classical example of laminar free convection, namely, at a vertical plate, the analysis of unsteady-state transport is very complicated. Only fairly recently have approximate and numerical solutions been given for the two basic unsteady-state boundary conditions: a concen- tration (temperature) step, or a flux step, applied at the plate surface. In Appendix E these solutions, insofar relevant forthe case of electrolytic solutions (Sc >> 1), are reviewed and compared with those few transition times reported in work on laminar free convection. . Table 4.1 lists some transition times calculated for this kind of free convection, following a concentration step in 0.05 M CuSO 4 solution. It can be seen that the transition time (to a flux 1% in excess of steady state) varies along the plate. Of course, in free convection the transi- tion time is also dependent on the density difference generated at the plate The dimensionless time variable in the transient process is (DgLp'l/2 T free cony. Ijx / (43) 166 Table 4.1. Transition.t•imes (in seconds) in laminar free and forced convection along a planar electrode, for a solution of 0.05 M.CuSO4, 1.5 M H2SO4 at 25 °C. Current densities indicated between parentheses (in ma/cm 2 ) Distance from 0.5 1 .5 10 40 leading edge (cm) . . Free convection at 42 60 133 189 377 vertical plate (2.84) (2.40) (1.61) (1.35) (0.95) Forced convection . . . between parallel . . . plates (1 cm distance) . . . . . . Re 10 95 151 . . .707 1777 (1.31) (1.04) (0.61) (0.48) (0.30) 100 21 33 96 151 382 (2.82) (2,24) (1.31) (1.04) (0.65) 1000 44 71 21 33 83 (6.08) (4.83) (2.82) (2.24) (1.41) 2000 28 44 13 21 52 (7.66) (6.08) (3.56) (2.82) (1.78) 167 The transition times in very dilute solutions, e.g., < 0.01 M CuSO4 , therefore become so long that serious difficulty is encountered in obtain- ing any steady-state value at all. It appears (see Appendix E) that this is particularly so where the transport process takes placein a restricted. space, but provides the driving force for a secondary convection pattern extending over larger distances. Examples are high-Sc free convection at a horizontal plate, 11 and laminar free convection in a narrow vertical slit which, at the top and the bottom, is open to a large reservoir of solution. 48 . Figure 4.3 shows an unusual limiting current curve obtained by Fenechfla in 0.01 M CuSQ4 solution at a horizontal electrode. The current maximum could be made smaller by applying a lower rate d(AE)/dt (2), but could not be made to disappear altogether. it is likely that the maximum is caused by the unsteady-state diffusion process close to the plate while the free convection pattern at larger distances is slowly being established. However, inlaminar free convection at vertical plates the existence of such an "overshoot", at least in high-Sc liquids, is dubious; if it • . exists, it is inconspicuous. 157,158 The risk inherent in long transition times, such as are characteristic of free convection with a feeble driving force, is two-fold: 1. The bulk solution may become depleted. In free-convection experiments the cathode and anode compartment are often separated by a diaphragm to prevent mutual interaction of the convection patterns; replenishment of the catholyte does not take place. Examples of sagginglimiting current plateaux due to bulk depletion can be found in the work of Boeffard 46 with ferri-/ferrocyanide in free convection at horizontal electrodes. c\r C-) 0 E 189 2. in copper deposition surface roughness may lead to an increase in the apparentlimiting current. This is particularly serious in potentio- static current control, whether through the applied potential or the working electrode potential. In free convection experiments, in particular, it is convenient tp apply a large negative potential and let the current come to its steady state value. As demonstrated in chapter 3, however, this may lead to a more serious roughness formation than if a series of current steps are taken. It is reasonable to believe that this finding is valid for free convective as well as forced cOnvective mass transfer. The second consideration is complicated by the fact that those solutions which have the smallest driving force and, therefore, the longest transition times, also have the lowest limiting current levels The surface roughness problem in free convection, therefore, can only be resolved by extensive experimental investigation. It is very probable that for any well-defined type of free convection there isa minimal surface roughness effect associated with each CuSO4 concentration. This effect could be incorporated in an effective diffusivity of the same kind as those determined in chapter 3 for the rotating disk electrode These diffusivities would then reflect a compromise between high apparent limiting currents due to unsteady-state free convection, and high apparen.t limiting currents due to surface roughness formation. 4.3. Forced convection Although the scarce experimental data mostly refer to unsteady-state effects in free convection, one should not infer from this that in forced 170 convection transition times are negligi.ble Recently solutions have been obtained for the flux response to a concentration step at a plate in 160,161 parallel flow 159 and in boundary layer flow (Appendix B). A solution for the analogous transient at a rotating disk has been given in Appendix B. Table 4.1 presents theoretical transition times following a concen- tration step at the wall in laminar flow between parallel plates .(Lvque mass transfer). Clearly, in slow laminar flow (Re 100 or lower) trans- ition times are comparable to those in free convection. Here, however, they are but weak1ydependent on the concentration, through the diffusivity; the dimensionless time-variable in unsteady-state mass transfer of the Lvque type is: T = —i (4.4) forced cony. t(~X 2 where S is the velocity gradient at the wall. The transition time is, however, more strongly dependent on the • distance from the leading edge (x) than in free convection at a vertical plate. I 0 4 0.28 M Glycerol 3.0 M Re 1800 1 mm n 1 min I mm E U . o. >- 5 min ( 5 min If) 5 mm C C 0. 10 mm 10 min 10 mm 0. 12.2 mm (TTT 12.2 min 12.2 mm 172. 4 4 Interaction with the potential distribution at elongated electrodes Two factors restrict the usefulness of the step-responsetransient solutions reviewed so far: 1 In most electrochemical measurements the limiting current is not imposed abruptly but approached gradually, controlling either the current or the electrode potential. Since the unsteady-state transport equation for forced convection is a linear equation, it is in principle possible to derive solutions for more complicated time-dependent boundary conditions from the step-response solution, by application of the superposition (D.uhamel) theorem. If the resPonse to a uniform step in the concentration, c 0 , at the electrode is i = i (x,t),.theningeneral: i(x,t) dt1 dx1 :tx(Xiti)i(XXiitti) (4.5) = f f Conversely, if the response to a uniform step in the current is c = then c(x,t) = f dt dx 1 (x,t i )c(x-xl ,t-t 1 ) (4 6) fo • : :Suppose the imposed boundary condition is simply a linear increase, st, of the current density, which is taken to be uniform everywhere, then the equation to be solved is: t 1/3 2/3 * D S (t-t 1 ). • c(x,t) = S dt1c0 (47) f (x-x•1 )2/3 • where the function c 0 (x,t) is a rather involved analytic expression, given 159 by Soliman and Chanibr. Note, however, that (7) implies that the current density is at all times uniformly dist.ributed;.this is obviouslyin 173 contradiction with the limiting current distribution which one would like to reach eventually. • In any case, for an application of the superposition theorem to mass transfer at electrodes with a non-uniform limiting current distribution, • it is necessary that the applied local current density is known throughout the approach to the limiting current 2. Mass transfer is not the only process controllin.g the unsteady-state behavior of electrode potential in respOnse to an imposed current, or vice versa. In particular in the copper deposition reaction an equation such as (6) cannot be used to represent the total imposed electrode potential interms of concentrations. The implications of this are illustrated by Figure 4 4, which shows the development of sectional limiting current curves recorded by Hickman 156 at a horizontal plate electrode in laminar forced convection. Here the total current is increased linearly, with time, and the potential of selected electrode segments is recorded. The limiting current plateau is reached first at the trailing edge of the plate, then in the center, and finally at the leading edge The time-difference is on the order of 4 minutes i e , approximately 1/3 of the time required to reach iL everywhere It is clear that if sectional limiting currents had been read with respect to the central overpotential, inaccurate local current values would have resulted for the edge segments If the total current had been recorded against the central overpotential, the correct total iL might have been read (due to compensating errors), but the plateau would not be well-defined 174 Figure 4.4, therefore, illustrates the practical importance of sectional measurements on extended electrodes, as first systematically la applied by Fenech) However, from the viewpoint of unsteady-state mass transfer the phenomenon observed raises several questions. The time in which the limiting current is approached (8-12 minutes) was selected by Hickman after preliminary experiments to establish a maximum rate of current increase beyond which the apparent limiting current would reflect unsteady-state.effects. Therefore, one would expect the measurement to be quasi-static, i.e., deviations from the steady state • are insignificant during the passage to the limiting current. Neverthe- less,.the limiting current is reached nonsimultaneously. The order in • which the apparent limiting currents are attained agrees, at least qualitatively, with the distribution of trans.ition-times.which follows from (4). In the unsteady state caused by a forced current increase, . • an apparent limiting current due to depletion at the electrode is reached first where the time to reach the steady-state concentration profile is longest. • • • • • • The time in which the limiting current is reached., i.e., the minimal time to obtain steady-state, is an order of magnitude larger than the transition times listed in Table.4.l for Re= 2000. A comparison can be: made in terms of the dimensionless time T (4), substituting fort the minimum time to reach steady-state, t ss = iL/(di/dt) for the trailing segment. This yields T = 2.7. The transition, time T 175 following a step in the flux at a surface in .Lévque type mass-transfer - is not exactly known; it is probably no more than three times the transition time following a concentration st.ep,which is 0.239 (Appendix E). The time of passage observed in the experiment of Figure 4.4 is an order of magnitude larger. To explain, first, the order in which the sectional limiting currents • are reached, one has to take into account that the applied current is neither distributed uniformly at all times (equation 7), nor similar to the limiting current distribution. If the steady-state is essentially maintained during the transition, the trailing edge will have a fairly high current density throughout (see Figure 2.14). This will lower the minimum time • required to establish a steady-state limiting current, compared to the • values suggested by Table 4.1. The distribution of this minimum time along the plate will probably be moreuniform than x2'3 as required by (4). Nevertheless, it is likely that in the final stage of the passage to limiting current, where mass transfer dominates, the potential distri- bution, each small increase of the current (now distributed approximately as'x 1/ 3) will require a local transition time distributed as x 2 " 3 . . At the same time, as Figure 2.14 shows, the steady-state reactant concentra- tion can reach a virtual zero level near the trailing 'edge.of the plate before the limiting current is reached. Therefore, even if, for the plate as a whole, the passage to the limiting current is quasi-static until very close to the limiting current (e.g., 95%), the attainment of the limiting current condition, i.e., cR 0, is very likely to be non- stationary and, therefore, nonsimultaneous. 176 If this explanation is correct, sectional limiting currents near the trailing edge of theplate will tend tobe somewhat higher than expected unless the passage to the limiting current is made very slowly, in particular near the limiting current. Note that no consideration is given to surface roughness formation and hydrogen development, which may reinforce the tendency to unsteady state. Clearly it would be •a formidable problem to obtain a quantitative picture of the transient current distribution corresponding to a certain rate of approach to the limiting current. For the present, it is sufficient to conclude that the minimum time allowable to reach steady- state limiting current by applying a current ramp, should be equal to, or greater than, that required for the trailing section. At this point the second question raised by Figure 4.4, needs an answer. The minimum time of passage to the limiting current, according to Figure 4.4 and equation 1, is an order of magnitude larger than the transition time following a step change in concentration or flux. It was thought desirable to check experimentally in a simpler sitüa- tion whether this particular relationship between minimum time of passage and step transition time is .a general feature of forced convective diffusion. The hydrodynainic model selected was the rotating disk; for this case numerical solutions are available both for a concentration step and for a flux step instantaneously applied. The transition following a concentration step is relevant for potential scanning. This was investigated as. an alternative way to approach the limiting current, closely related to linear increase of applied 177 potential in well-supported solutions To avoid interference of electrode kinetics with the transient mass transfer, the electrode reaction was reduction of ferricyanide at a nickel electrode in equimolar K 3Fe(CN) 6 + K4Fe(CN) 6 solution with excess NaOH From the few investigations 162 ' 3 available in the literature one can conclude that the minimum scan period necessary to reach a steady- state limiting current is of the same order of magnitude as the step transition time. There appears to be, therefore, a significant difference between the passage to limiting current potentiostatically as opposed to galvano- statically This should be observed most clearly at a rotating disk limiting electrode, where the/current distribution and the transition time are not distance dependent. . . Part II Unsteady-state mass transfer at a rotating disk electrode 4.5. Experimental The experimental equipment is essentially the same as that described in chapter 3 for the potentiostatic measurement of rotating disk diffusivities The Wenking potentiostat was used both in the potentiostatic and galvano- static experiments, in the latter the potential drop over a fixed resistance was controlled Current and electrode potential were recorded by a Vidar 12289 Low- Level Data Logging System, consisting of a idar 510 Integrating Digital voltmeter, Vidar 12029 scanner, Vidar 625 Digital clock, Franklin 1040 High-Speed Printer, and vidar 650-8 Coupler. The data in digital form were 178 Table 4 2 Experimental solutions = C CNaOH = 0.4 M Ga;lvanostatic measurements. c 3 =0.0l84M Fe (CN) 6 p = 1.022 g/ml p = 0.985 cp D = 7.88 x 10 6 cm2/sec • Sc 1223 ; pD/T = 2.0x de/ ° K Potentiostatic measurements c =00143M Fe(CN) p = 1.021 g/ml = 0.984 cp -6 2 D=7.89xl0 cmsec 10 Sc = 1222 , pD/T = 2.60x 10 dyne/ ° K Note 1 D calculated from the limiting current by means of the Levich equation. 2. According to Gordon, Newman and Tobias 47 pD/I. = 2.35 x 10 10dyne/ 0 K; according to Arva, Marchiano and Podesta51 2 52xl0dyne/°K0 179 either printed out directly, or stored on magnetic tape and processed by computer. The highest scanning speed was 30 signals per second. The rotating disks employed were of the design shown in Figure 3 1, with an embedded nickel electrode Before each series of experiments the disk surface was pre-treated by cathodic hydrogen evolution in 5% NaOH solution at a current density of 20 ma/cm 2 for 10 to 15 minutes, to obtain a stabilized surface. The nickel reference electrode was given the same treatment. The electrolyte was an equimolar solution of K 3Fe(CN) 6 and K4Fe(CN) 6 prepared from reagent grade chemicals, with NaOH added as supporting electrolyte The concentration of K 3Fe(CN) 6 was approximately 0 02M, that of NaOH 0.4 M. The exact reactant concentrations and physical properties for the two series of measurements are given in Table 4.2. Theferro.- cyanide and NaOH concentration were made up to the indicated values but not analyzed Experiments were conducted at 25.0 + 6.1 ° C 4.6. Results A. Galvanostatic current ramps At,two different rotation speeds (115. and 491 rpm) a series of current ramps were applied, with di/dt ranging from 0.005 to 12 ma/cm 2sec Figures 4.5 to 4.7 illustrate the curves obtained The individual points are overpotential readings taken at 60 msec intervals The limiting current plateau gradually takes the character of an inflection, as di/dt is increased The apparent limiting current was taken to be the value at the inflection point, this was located at approximately -900 my overpotential except at the highest increase rates, where it tended to shift to higher potentials, up to-1200 my • :. . . .181. tIOO,iyi . V0l.tA4t4S. TINE ZN CN*N. 1 SELV 10-IS. • • . . . . • -• ..-N0. 22 • . . . I4O* . . . . . . • • . .. a • . . . . . :- 1200 .. . . . • . . . . .. - • . . . •. • . • . .. .. -, .• .' •. . . • . . - . . . . ..I0oo. •• . . . •• . . . p . .. . . 000*- . . • . E . : E . .. ,.• •. . •..-.• 4001 1 200 0• • . . . . 4 4' • 4 • .0 ••• . 0 :0 :0:. e. - .0 0 •• ,. . . . •. . . • S S. XBL 716-6883 Fig 4 6 Overpotential (my) vs time (sec) for a current application rate of 0.266 ma/cm2 ec (exp 21) 182 '•O• ..• _._I ,s rn # t.tWUS. t its VCW*. S .. V S-$ . . SUNNO.. V till I '1 I 100.0 • V V ••. •. - • • V V • 0 ••• • • • V • • • 4 V 4 • • • - • ••• • • V V••V V.. -- •• V •. - •- V ••• - • - • • • • • • - - • •• • • • V •• V • • • • • • • • • V . • • • I I 4OO • 1•' . I '- • .4. • .4 .4 4'. • • .' •-' .'.-..•' • V V V • -- V • V V XBL 716-6884 Fig Overpotential (mv) vs current x 500 Q (mv) for a current applica- 4.7. • V V • • . •. • -.tion rate.of 2.04 m7crn2sec at 115 rpm. (éVXp.V 28). •• 183 Table 4 3 lists the results and gives also the values of dimension- less quantities, in terms of which the results at different speeds can be compared The time in which the limiting current is reached,i/(di/cjt), is represented as a Fourier number with respect to the diffusion layer thickness: -1/3 T = Dt/6 2 tDV/v3 = . (4.8) This dimensionless time is the same one used to describe the decay of the flux following a concentration step, at the rotating disk surface, or the concentration response to a constant flux instantaneously applied at the surface. (Appendix .B and E use T = 0.307 .wt Sc *' 3 ). The minimum time to obtain steady-state limiting currents by means of a current ramp is apparently an order of magnitude larger than = wt SC = 5.46 , (4.9) i.e., the time in which the steady-state concentration is established (to within 1%) following a current step at the disk. Figure 4.8 shows the results in a double-logarithmic plot of apparent limiting currents, relative to the steady-state value,.against dimensionless time (8).. At short times the apparent limiting current appears toterid toward a cube root dependence on time The steady-state limiting current values at the two speeds were in good agreement with the values for integral diffusivities of ferricyanide 47 reported by Arvia a o (Table 4.2). Table 4.3. Apparent limiting current values obtained by linear current increase. A. .115rpm. di/dtxlO 3 i 1L/13 2 (ma/cm sec) (ma/cm ) 16 . 5.03 3.18 723 0.985 15 543 316 670 0978 13 103 323 353 1000 14 . . . 10.4 . . 3.23 350 .1.000 9 217 323 168 1000 17 816 337 446 1043 20 163 3.56 22.3 1.102 18 172 338 211 1046 22 261 3.60 13.9 1.115 21 266 362 137 1121 24 494 400 736 1238 23 506 403 719 1248 26 1063 . . 4.57 . . 3.42 1.415 25 1150 . 4.62 3.16 . 1.430 28 2040 5 23 1.78 1 619 27 2140 5.35 1.70 1.656 29 4360 6.42 0.834 1.988 30 4680 6.48 0.777 2.006 31. . 6975 7.66 0.522 2.372 32 .7040 7.62 0.517 2.359 185 Table 4 3 (continued) • B.49lrpm di/dtxlO 3 i WSc "3/(di/dt) (ma/crn 2 sec) (má/cm) • 30 130 5.19 192 1 34 133 5.19 188 1 36 258 5.20 96.7 1.002 35 • 260 5.24 • • 95.9 • 1.010 37 414 531 602 1023 38 421 531 593 1023 39 835 553 299 1065 40 857 548 291 1056 41 1700 5.96 14.7 1.148 42 1760 5 71 14.2 1 100 43 3370 • 63S 7.41 1.223 44 3400 6.29 7 34 1.212 45 6660 7 30 3.75 1.407. 46. .67.00 . • • 7.42 .3.73 • 1.430 . 471 11620 8.34. 2.15 1 607 48 • 11620 • . ••• : .42. 215 • • .1.'622 • 187 B. Potentiostatic current increase A series of potential scans were applied at 490 rpm, ranging from d/dt = 5 mv/sec to 2 v/sec. Figures 4.9 to 4.12 illustrate the curves obtained. Note that the potential scan did not start from exactly zero my overpotential but from a small negative bias potential, so that the current shows a step at time zero. Characteristic for potential control in the limiting current region are the current fluctuations (+ 1% • standard deviation) with a frequency Eomewhat higher than 1 sec'. The limiting current value was determined by averaging over the plateau, which here extended from 400-800 my. At higher scan rates the plateau is reached at lower values (< 200 mv); • this is apparently due to a.current maximum at lowpotentials, which at the highest scan rates can be clearly recognized as such. Where the current peak is higher than the plateau, the latter tends to be in unsteady-state, i.à.., its current value is higher than the steady-state limiting current. At the highest scan rates a saddle-type curve is formed (Fig. 4.12). Table 4.4 lists the peak and plateau currents, and the potential scan times. The latter can be expressed as • • • -1/3 • -1/3 w Sc - Sc -1/3RT • T - Wt C • d4/dt - (dri/dt) nF • (4.10) The time of passage necessary to obtain steady-state limiting currents by potential scanning is apparently of the same order of magnitude. as the transition time (to within 1% of steady state) following a concentration step; - • • T = h/3 = 2.05 • (4.11) (see Appendix B) 189 I - I- b:CWAN. 1 Vs. CN*M. 1 . . $I&.V i-i.'. io,is,s•. - *UNNO U • . •• •. ••.• : e •• . . . . . . . . . a H .. . . . . ,• . . . . . . .4 se .4 ••• : S a.4 I 4.. 4 . .4 •••. •,. . . . . •• 4 • •. . . I • . . . 4 *IIII 14U#4 $41441141444 I.414$44A$ U4$u*4L* I III II 1% II 3 ' AMP p a a XBL 716-6886 Fig 4 10 Overpotential (my) vs current (ma x 500 Q) for a potential scan rate of 27 9 mv/sec at 490 rpm (exp 62) 190 A . ,,,, Si, Si iIItSSiS SI ii IIIrlItIvtl.v1.vr.IItItp ,,,,,l,,I,,,I,,S,9,,,,,I,,.,,(,,,,,,.., • t vS. .. . HAP4. 2 . $CLV tOtS. 10/10/60 - RUNNO SI V. • . .. V . V • . V . , •• •V . • V ( ¶000- V V 000- V VV V •• V . I . . .t , . V600 400: 200- 0- 0 . V . -. .-200- . • V V 0 0 0 0 0 0 0 0 0 0 I 4 II S 0 XBL 716-1030 Fig 4 11 Overpotential (my) vs current (maxSOO ) for a potential scan rate of 114 mv/sec at 490 rpm (exp 64) 191 • . - . . l4 OOIIIIiI 1 1r1ir,-, -Ir,y..., CHAN. 1 : V$. CHAN. P $I.V iO-i•, 10I,•I$•. •U$*O 70 t000.. . . - . .. . . . . 4 a • . •00'- . . . . . • .5 . ... . . a . 4 • . .: .. . . . . . .. . .4 0. • . . o06.. . . . . . S . . •. . .. . 4 4 40O. . . . . . . . . .. . . . 0 206 . 4 e. 4 . .5 . . . . . . S • . S . . .. . . S S ha 0 • .. -. -.:-- -. • . •. . . 0 . • - - a • XBL 716 7 6887 Fig 4 12 Overpotentialt (mv) vs current (ma x 500 ) for a potential scan r a e of 436 mv/sec at 490 rpm (exp 66) 192 Table 4 4 Apparent limiting current values obtained by linear potential decrease (490 rpm) dr/dt wSc 1/3(d/dt) iL 'max 1L/1 max (mv/sec) (ma/cm 2 ) (ma/cm 2 ) 58 4 55 4 76 - 26 50 0.996 - 59 877 478 60 176 479 - 685 1002 - 61 27 9 4.79 - 4 33 1 002 - 62 27.9 . .4.80 . - 4•3 1.004. 63 547 484 - 221 1013 - 64 114 4 89 5.16 1 05 1.023 1 079 65 221 5.01 S 95 0 546 1 048 1 245 66 433 5 14 8.30 0 279 1 075 1.732 67 436 5 13 8.28 0 277 1 073 1 736 68 633 5.46 9.48 0a,919 1.142 1.983 69 920 5.70 11.05 0 131 1 192 2.312 70 2165 8 06 16.5 0 0557 1 686 3 46 71 2097 8 06 16.5 0.0575. 1 686 3.46 194 •Figure 4.13 shows the results in a double-logarithmic plot of apparent limiting currents (plateaux) and peak currents, relative to the steady-state limiting current, against the dimensionless time(lO). At short times the apparent limiting current appears to depend inversely on the square. root of the dimensionless time. The steady-state limiting current was again as expected on the basis • i 51 of d ffusivities given by Arvia a.o. (Table 4.2). 4.7. Discussion The results of both galvanostatic and potentiostatic unsteady-state experiments may be interpreted in terms of a pure-diffusion model which is the more valid as the current or potential increase rate is higher. Near the disk surface the convection velocity toward the disk is • v = -y2w312 v' 2 • • • (4.12) y where. a= 0.51023.. The convective term in the equation of convective diffusion is very small if • • • • ± << 1 y << 2Sc 3 • (4.13) i.e., an order of magnitude smaller than the steady-state diffusion layer. In very fast transients the limiting current condition (c 0), or a current maximum, is reached before the diffusion-layer thickness has • grown enough to make convective transport important. This is evident in particular from the potentiostatic experiments, where thepurely diffusive phase distinctly precedes the convective-diffusion. 195 phase, if the scan rate is fast enough. . . In the followinganalysis, to facilitate interpretation, the pure diffusion model is first applied to a deposi.tion reaction. The results are then reconsidered for a redox reaction. . . . . A. Galvanostatic experiments . .. .... .. . . . . . . The response of the concentration at. the electrode, c, to a linearly increasing, flux:. . . . . . N = -D(-)y. 9 = +cst (4 14) 164 . . . . is - : .. .. . . . .. ,'. c -c 3/2 o b_ 4st Cb - 3(iD) 1' If the flux is toward the electrode (N < 0) and = IdN/dtl , (4.16). cb . , ... zero concentration at the electrode is reached in a time =( ' (4 17) \16 s J and the flux will be 1.209 cb(sD) 1/3 Therefore, at high current increase rates the apparent limiting current should be proportional to (di/dt) 1 " 3 In terms of the dimensionless variables of Figure 4 8 196 1 —l/3\—l/3 (lcjA)SC i /iOO = 1.662 . (4.18) L di/dt ) This equation is shown as a solid line in Figure 4.8; the experimental values are definitely higher but tend toward the theoretical line at higher values of di/dt (or w). It is easy to see that the deviation from (18) can not be ascribed to the, fact that a redox reaction is involved. Since the flux is controlled, equations 14-17 are valid for the oxidized species, i.e., the ferricyanide ion (ox); the accumulation of the reduced species (red) at the cathode will be according to (15) but with D substituted for D, and s equal to red ' ox red s0 in an equimolar solution.. Calling /16 s 2\1'3 .. ' ox rg = t 9.n.D) . ' ' (4.19) one has for the concentration overpotential: - RI l—T312 ' LT (4.20) Ti - F 3'2 i+RT '' g where . R = (c D 2 )/(c ox,b ox red,b Dred .1/2) , 4.2l) For the present solution R 1.1. From (20) it.is clear that the time passed until the potential is sufficiently negative for hydrogen' ion' ' reduction, will be less than the theoreticaltime (17), i.e., = 1. The Tg limiting current then ought to be less than' (18).. A similar reasoning shows that charge transfer overpotential, if influential, would lead to a lower limiting current. The higher limiting current value, therefore, should be ascribed to convectjon. In the final phase of the transition, where the concentration ofreactant at the electrode is exponentially, decreasing, convection becomes important and prolongs the transition, thereby increasing the current. It is of interest to compare the unsteady state currents found by Hickman (Figure 4.2) with equation (17). The proportionality between .t and s 21 is very satisfactory (Figure 4.14). The diffusivity of Cu derived from the slope is D = 4.73 x 10 6 cm2 /sec; this is in good agree- ment with the values reported in Chapter 3 Note that the point for lowest di/dt deviates most from the least-squares slope through the points of Figure 4 14, this is probably due to secondary convection taking place at the edges of the embedded electrode. Experiments similar to those of Figure 4.2, but for ferricyanide reduction ata nickel electrode, were reported by 13offard,16 who also established the dependence of the apparent limiting current on (di/dt) '3 . B. Potentiostatic experiments The response of the flux at the electrode, N, to a concentration step, Cb_CoI is well-known: - N= -D(-) = (cbc) (4 22) If the surface overpotential or ohmic potential is negligible, one has from the Nernst relation: 1.0 -a 0 .'-. o C 0 4- 0 4- C C.) C 0 202 while the potential at. the peak should be •independent of dn/dt: = -0.8540 R T . . (4.29) max . 0 It is to be noted that the peak current is not a limiting current in the usual sense, since Cb = 0.426 c. Figure 4.16 illustrates schematically the sequence of concentration profiles, assuming a Nernst layer expanding proportional to /t and the concentration at the electrode decreasing as (2/3) t . . In terms of the dimensionless variables of Figure 4.13 equation 28 is: -. I\d4/dt When a redox reaction is involved, one should account for the C0fl tribution of the product ion (reduced species) to the overpotential. The accumulation of the product ion tends to decrease the potential to more negative values; when the potentia.l is controlled, the result is a lowering of the current. The effect will be negligible in solutions with . a small ratio of reactant (ox) to product (red) ion bulk concentration; in that case the conversion rate at the cathode is too small to cause a large shift of the potential with respect to the equilibrium potential. The effect will be large in solutions with an excess of reactant (ox) ion; a mathematical solution is available for this case (Sevcik-Randles equation). . . . . . . . . 203 In the present case . of an equimolar solution no simplifications are possible. Consequently, a numerical solution had to be obtained using the characteristic diffusion parameter tc \/D \1/2 R = \Credbt\b 1tD red! ) . (433) • which for the equimolar ferricyanide-ferrocyanide solution .is approximately 1.1. Appendix F gives the details of the numerical solution, which employed • a procedure given by Acrivos and Chambr) 69 As a check on the accuracy of the solution, the parameter R was varied from l0 to 10 by decades. Figure 4.17 shows the current, expressed in terms of: - •_____ I • I . 4 4 • c nF(aD ' ( . • . • . ox,b oxj1 against the logarithm of the dimensionless potential 4, = -nnF/(RT). Below R = • 10 the current maximum has the value given by (28), as in a deposition reaction, and its potential is given by (29). At high values of R, i.e., for potentials much more positive than the standard redox potential, the current maximum has the value given bythe Sevcik-Randles equation 0.4463 • ' ( 4.35) • m'oo cOX .bnFaDOX • where the constant agrees to 3 significant figures with that computed .by Nicholson .and Sham. 166 The peak potentials are dependent on R, shifting • • by -Ln 10 for every decade• increase of • R. • • • • I.0 C cu o. 0 205 The current maximum for the equimolar solution is: • i 0.52738 c nF/aD , (4.36) • max b ox or 1w Sc3\ -1/2 • 1max"c 0.84995d,d (437) ) The peak potential is 4 = 1.6, or n = - 41.1 my at 25 ° C. Figure 4.13 shows that, for fast scans, (36) predicts the apparent limiting current reasonably well. The minimum scan period.to reach steady- state limiting currents, is: T - d4/dt. -1.40 (4.38) Note that this is somewhat shorter than the.transjtion time following a concentration step at the rotating disk (T = 2.05 for i/i 1.01) Girina, Filinovskii and Feoktistov 162 measured current maxima for reduction of Tf. at copper amalgam, by potential scanning at a rotating disk electrode. They found good agreement with (35),whiçh is the appropriate equation for, this reaction. As expected, the minimum scan period is accord- ingly shorter; the reported value corresponds to t = 0.86. The experimental peak potentials were larger than predicted, ranging from 65 to 190 my, and increasing with the scan rate. The ohmic drop - included in the measured overpotential is probably responsible for this shift. The resistance of the solution between cathode and reference electrode was estimated at 15 0. At the highest peak currents this.leads to an included ohmic drop of 35 my. It is not unlikely that the inclusion 206 of.. ohmic drop in the controlled potential leads to a shift of the peak potentials as large as .the one, found experimentally. In view of the good agreement between predicted and observed current values this possibility was not explored by further computations C. Minimum time to reach steady-state limiting current The minimum time observed in potentiostatic scanning is an order of magnitude smaller than that in galvanostatic passage to the limiting current. This can be explained by comparing the current-time relationship in both. In ,galvanostatic scanning this is linear. The current during the initial stage of the passage to limiting current is low , while the mass transfer process is still purely diffusive. When convection becomes important and starts to retard the transition to a steady-state 'concentration profile, the diffusion layer is not very far depleted and, therefore, the retarding influence of convection is effective. On the other hand, in potentiostatic scanning the initial current is higher. This is indicated schematicállyin Figure 4.18, where the time is scaled with respect to the minimal time of passage to.steady state (t 5 ). 'The potentiostatic curve is that given by (30),. i.e., for pure diffusion but in a restricted layer (Nernst thick.ness); the steady- state coincides with the maximum. Here the retarding effect of convection is less since the solution at the 'electrode has been depleted to a larger extent during the early stages of the transition. In the pure diffusion model, on which equations (18) and (36) are based, this difference in current profiles leads to a difference in t ss expressed in the variable T(9) the galvanostatic criterion is Tss = the potentiostatic one is T 0 85 When convection enters the picture, ss the difference is reinforced according to present results the galvano- . 2. static criterion, is t . 60, the potentiostatic one ss The min.imuinpotentiostatic.scan time is practically the same as the transition time following a concentration step. The minimum galv.ano- static scan time is approximately 10 times the transition time following a current step This last result confirms the conclusion inferred from the experiment of Figure 4.4 4.8. Conclusions 1. Theoretical information about transition times following a concentra- tion or flux step in forced convective diffusion can be used to estimate the minimum time necessary to establish a steady-state limiting current by means of a current ramp or potential scan In the potentiostatic mode this time is of the same order as the transition time following •a concentration step However, in the galvanostatic mode the minimal time req!.4red is an order of magnitude larger than the transition time following a flux step 2 Experimentally the minimum time to establish steady-state at a rotating disk was found to be 2 sc1'3/w in potentiostatic scanning, and approximately 60 Sc 1" 3 /W for current ramp. These values are valid for a redox reaction in equimolar solution, with the assumption of very fast electrode kinetics 3. The apparent limitIng currents generated by fast potential scans or current ramps at a rotating disk electrode can be satisfactorily explained by a pure diffusion model The agreement is good for the current maxima observed in potential scanning. . Qualitative agreement is found. for current ramps, where-the apparent limiting current is influenced more effectively by convection. . 210 CONCLUSIONS AND RECOMMENDATIONS • 1. Properly designed and executed limiting current measurements represent • a powerful tool in the study of transport phenomena in forced and free convection at solid-liquid boundaries. This is illustrated by the continually expanding use of the method in chemical engineering, reviewed in Chapter 1. Accurate measurement of.limiting currents requires first of all well-defined limiting current platèau. It has been shown how an ideal plateau for a given model reaction can be constructed. On this basis the limitations inherent in the copper deposition reaction, as compared to the ferricyanide-ferrocyanide redox reaction, can be distinctly • recognized. The slow electrode kinetics lead to a loss in the length of the plateau; the surface area change due to deposit roughness causes an inclination of the plateau when high current levels are involved, or when the steady-state limiting current is attained very slowly. In spite of these shortcomings the copper deposition reaction remains in frequent use, especially in studies of free convection. Correct interpretation of limiting Current values requires knowledge of the migration contribution included in the limiting current For the copper deposition reaction the migration contribution is strongly dependent on the degree to which the HSO ions in solution are dissociated. Calcula- • tions indicate a migration effect roughly twice that for complete dissocia- tion of HSO 4 . To correlate mass transfer rates from.limiting current measurements ionic, not molecular, diffusivities should be used. The effective ionic 211 diffusivity of Cu ion in acidified CuSO 4 solution at 25 °C, determined at a rotating disk electrode, can be expressed as -10 MD/T (2.01. + 0.13) x 10 .dyne/ ° K and is not significantly dependent on the ionic strength, at least in solutions sufficiently dilute in CuSO 4 (e.g., less than 0.1 M). This is substantially lower than the molecular diffusi.vities which have long been in use for this system. In more concentrate1 solutions the ionic diffusjvjtjes obtained reflect an increase of effective surface area due to deposit roughness. The product pD is clearly dependent on the CuSO 4 concentration when the latter exceeds 0.1 M. The values obtained by agalvanostatic, stepwise, approach to the limiting current are apparently less influenced by this effect than those obtained by potentiostatic measurements, even if a potential-step method is applied. By solving the time-dependent equation of convective diffusion it has been shown that the current following a concentration step at the surface of a rotating disk decays to a value only 1% above the steady-state value within .2.05 Sc seconds. The transient currents following a potential step to -500 my at a rotating disk electrode need a time approxi-. mately twice as long, probably due to rearrangement of the deposit on the disk. . . . . . . The nonsimultaneous development of local limiting currents at elongated. electrodes is difficult to avoid, even if the time to reach the limiting / current has been chosen, so as to accomodate the most slowly reacting segment, e.g., at the trailing edge of a planar electrode. in forced convection. 212 This is a result of the interaction of transient mass transfer with the current distribution below the limiting current The latter may, • even in the steady-state, create a virtual zero concentration at the last electrode segment. 8 The minimum time to reach a steady-state limiting current is an order Of magnitude longer in a galvanostatic approach than in a potentiostatic approach to the limiting current. This was demonstrated in Chapter 4 by measurements at a rotating diskelectrode, using a redox reaction. This .conclusion has the interesting implication for the copper deposition reaction that the minimum roughness formation inherent in a galvanostatic approach to the limiting current may not be much less than that in a potentiostatic scan, using the maximum allowable scan rate to •reach the steady state. limiting current This could be confirme4 by performing experiments of the same kind as those of Chapter 4, using the copper deposition system. A quantitative characterization of the roughness formation by means of impedance measurement would be desir3ble 213 ACKNOWLEDGEMENTS I would like to express my gratitude to Pr.ofessor Charles W. • Tobias for his personal interest and encouragement throughout the course of this work. Where the thesis tended to become a vast microcosm, his intuition and practical experience were a frequent reminder of the world outside and the road ahead. I am also very grateful to Professor.John Newman for his constant attention and patient help in the analysis and interpretation of many re- sults. His physical insight and critical perception have made it enjoyable to spend so much time in what often appear to be the dry places of the mind. lam grateful to Professor Douglas Fuerstenau for reviewing this manuscript. . . During the course of this work I have had support and sympathy from many individuals. Here I would like to thank in particular Uziel Landau and Samuel Kiprono Ar.apkoske for their cooperation in the experimental work. I would like to thank Mrs. Jacqueline CrOss for graciously typing the extensive manuscript,, and .Gloria Pelatowski for the work involve4 in the drawings. Financial support was provided by the Atomic Energy Commission, through the Inorganic Materials Research Division of the Lawrence Radiation Laboratory. 214 Appendix A. Program for calculation of a theoretical limiting current curve.. . •. . Input: Z(Z), TP(t), XM(M), XNP(N'), ALPHA(ct), BETA(S), GA4vtA('y) The program puts F(f) = 0.01 and calculates the concentration over- potential ETAC according to (2.33), the ohmic contribution PHIOHM (A(P according to (2.37), and the quantity . •A E -Mf/(l-f) 1 Aninitial guess forE() is made, assuming that, cathodic surface over- potential constitutes the total n. Then the 5 -dependent part, of (2.34): AOEe -e S . .. . is compared with A, and if they differ by more than 1 in 10 6 , a new E is calculated, using DA dA0/d 5 = 4(k) A-AQ(k) After convergence ETAS(n5), V(total overpotential) and R(flc/fl) are calculated 215 LT''(rP'JT,'rJT) t# f?'44r(4F,.)) 'AF1 1'r FJAT 1X5HZ 1T= F l'?5 93'T4-,i 367i, 1)6 .nri x 9 3 ,I F , a- v tR-4 ?AT) ') 117 1 ,41 119,TP,X9 XND ('•r)-)) T 1; 112 0 T'jT 1)7 3 7 1I1 i')49Z,x1.xNP.r °RI'IT 1 1,, aLP4,1ra,A\4 \, 4 ° ,RjlT hf, r,o 1 1019 F (T.L.)) () 1') 7 •; r- i) 7 =+r.ii A- X F,(1.-F)c1•.1, ? Jzj)) - iF ()S(\_4:)).LT.1.S ()) D TD LDHA(A LPA* R1\X?(_FTA*) 2 r:._- .•1 )56) ?*/Z VTAS +1-.V7 .4.D . Tr)L4A 1.D2TIT • 216 Appendix B. Flux at a rotating disk after a concentration step at the • surface, in the limit of high Schmidt numbers. 1. The equation of convective diffusion, including a time-dependent term: • - c a + V -- = D • (B.• 1) -p.4 • , is to be solved with the boundary-conditions c=O , y=O , t>O ;. (B.2a) -00 c=l , y , t>O ; (B.2b) and the initial condition: c=l , y>O , t=O (B.2c) • In the limit of high Sc numbers the velocit.y perpendicular to the • disk can be set equal to: 3 / 2 _ 1122 , •= = 0.51023 • . (8.3) * Using the dimensionless variables: : T = tw(/3) 21 Sc' ; (B.4a) = yw2v12(/3)1/3Sc1/3 ; (B.4b) • the equation and boundary conditions can • • Note that t in this Appendix is (ç/3)2/3 = 0.307 times the variable, T = t sc 4 " 3 used in Chapters 3 and 4. • 218 = l/1'(4/3) = 1 1198465 (B 10) • 3. Equation (5) can be written in termsofa new vaxiable embodying the short-time asymptotic behavior = Z fl/2T 11 (B 11) • The result is: - (+ 6ZT) = (B 12) 4az with the boundary conditions c=0 , Z=0 , T>0 , (B13a) c=1 , Z- , T>0 (B13b) • • The i:nitial condition is now implied in one of the bowdary conditions. A perturbation expansion about the short time solution: c(Z,T) = c(Z) + T 312 C 1 (Z) + T 3 C 2 (Z) + , (B 14) yields for the zero-order term • dc 2 = Co) i e , C0 = erf Z, in agreement with (7), and next jd,2cl T112 dc1 2 dc + 2Z - 6c 1 = -24Z dZ2 7 219 One solution of the homogeneous equation is The complete solution is + 3-1/2 z22 = (Z 3 + Z) erfc Z The second-order term in (14)., obtained by o1ving: • .• dc dc dc dz . 2 + 2Z12c 2 = 24Z2 , . . .. (B.19) ••. : . . . . is 2 c = - - Ze 2 Z6erfc Z - 3Z4erfc z (52Z4 - 8Z2 -3) (B 20) l0V For the concentration gradient at the electrode this yields 3 + •3j- = +T3"2_a + O(t9" 2 ) . . . . (B.2l) •• oZ •o .2 •,,ç• . . .•• . . •• . .• . . . . . or, . ...... 51. 3T . 3T 2 .•. . . . . . . + .. + o(T ) . . (B.22) •° "F 2O1. •• . : . •• •.• .... 4 A numerical solution of (5) can now be sought using two sets of variables: . . . •• . . . . .. .• . . •: A. At times T < 1 the variables are 3/2 T = Z = fl/2t 2 = n/2T3 , (B 23b) 91 (Z,T) = c(Z,T) - erf(Z) (B 23c) S. The numericaiprocedure is the samefor both equations (24and(26). 221 Let the concentration ce1 , resp 2 ) at step i of T and at step j of the distance (Z, resp n) be called C 13 The left-hand side of(18), resp • (20), is eva1uated..as a central difference. The right-hand side, averaged between steps, is the central value of the left-hand side E g , for :. .. .. • (24): . : . .. . • (C. . +c. • . . . i+1,j i,j lj 1+l,)+1 1+1,3-1.. -2c . :.T.12Ti h . + c11 + c1,31 - 2c1 + 2Z 2) + ( + C1)1 2h 2h + (B 28) HereT=!(T +T • • . •• ) ... 2 i •i+1.. . ••• • . This results in a set of equations for the unknown concentrations • at.timé-step i. •+ .1; • • • • • • • A.c. . +• . + Dcc. . =: .G: • • • :j • • (B.29) i+l.,j-1 . .j.i+1,j j r+1.,j+l • • • • • • • The coefficients for the case of (24) are * • . • . A = -AT/12h 2T + ZT/12hT + Z2T/h , (B 30a) B = 1 + T/6h2T D = -T/l2h2T - ZLT/12hT - Z 2tT/h (B 30c) *. • • • • ., • .. .• • • The suffix j of Z is omitted hereafter • ...... 3. ,. ..• • •,• 222 G. = c. + AT(c. + c - 2c. .)/12h 2 T 3 1,3 i,j+l i,j-1 1 , 3 1 )/2h + T(Z/6T + 2Z2) (c +1 - c (B 30d) • 2 +T8Z 2 e AT * Using the relation : c = + Ec 1 , (B 31) we have from (23) • A.. + A.E. . + A.E. E.c. 3 3-1 3 j-1 j 3 j-1 3 j+l + B.E. + B.E. c + D. c = G. (B.32) 3 3 3 3 j +l 3 j+l 3 and from this = (G-A 1 )/(B + AE 1 ) , (B 33a) E = -D/(AE + B) (B 34b) 1 The successive Ej,,E are calculated starting with = 0, E 0, i.e., the boundary. condition 0 = 0. The term G. for the first time-step is zero. The calculation of c is performed in reverse order, starting with8=0,.00 • The derivative at the interface is calculated at every time-step: (4 C 12 - c 1 3 )/2h + 0(h 2 ) (B 35) A listing of the program CONST is given at the end of this appendix By variation of mesh size (h) at constant number of meshpoints (201) * The suffix i+l is omitted where the unknown c is meant. 223 o . .• (7) 00 0) - 'O 1- '0 0) t) t') LI) (N 'd- O 0 N -4 1.4 4è LI) (N N ) '0 ItT U) 00 0 (N 0 00 t) N L/) (N V) 00 0) 00 14 U) 4 (N LI) II) '0 C) U) 0 00 C) '0 o LI) C14 N LI) V) (N o ) N J) t) (N •-1 44) • E 0) U) (4) - (4) 0) 1q, N 0 U) 14) 44) le 00 0) $- GO - (N 0) '- LI) 00 0) 00 0 LI) -4. '.0 44) 0 o (4) 0) (4) - 00 L/) -4 U) 0 0) P4 N (N (4) (N r4 4.) ...4 r4. (4) - .4 '0 '0 LI) (N (N 4 .4 Cd 0) NI '0 LI) NI •.. N LI) (4) CN 0) (1) LI) . (4) 0 (N N I.f) t.. . . (4) U) (4) 4 U) ('9 0) . o • . '.0 (N 0) qdl N 4/) '0 (4) 0) - (4) (N cr . 0) lqT 00 •r-4 (4 00 .4 0) 00 U) -4. U) N .-4 N t (4) C) "? 0) (4) -4 r-4 '.0 (4) '.0 '.0 (4) (4) 4 4 '.0 (N '.0 LI) (N N LI) 1 4/) (4) (N ,-.4 -4 U). -4Cd U) U) 0 (N N LI) N -4. r (4•) LI) (N .-4 (7) c'1 14) E i- N LI) '.0 0) -4 1) (4) .N LI) 14) NI 0) 00 -4 U) -4 . (N LI) (4) N - -4 0) N 00 0) Or • 0) -4 (4) -4 - '0 (4) '0 '.0 (4) (4) • ... • • •. -4 '0. (N '0. U) (N N U) (4) (N •4 4 4 ..4 (N (N N LI) 44) (N 0) U , Cd 0 ' LI) -4 (4) 0) N 4/) If) '0 •• (4) C) CO • E 0) 't N L/) 00 -4 0) 00 14 LI) 00 U) (N C) -i 00 . 1-4 . 0) 00 . If) C) •' 0) 14 (4) -4 ' (N 0) 00 0) 0 0) (1) . ,-4 .-4 '.0 (N '0 11) (N 14 00 N N - '.0 (N . '.0 U) (N N- If) (4). (-4 - -4 0 0 0 0) - N If) N) (N 4. Cd 0 h. C) : (N N U). N - 44) '.0 U) ('9 U) 00 . 4) .00 (4) . 0) ' N LI) '.0 . (4) 0) 1-4 (4) 44') N LI) '0 1-4 0) . 0)1 00 . 0) 00 - L/) .-4 , (N L/) (4) N -4 00 CO C) . J• 0) -4 44) - - '.0 (4) '0 . '.0 (4) (1) _4 - -4 .10 • . • • . • . . . .•. • Cd - '0 NI '.0 $4 . N LI) •(N - - 0 . -4 -4 1-4 -4 -4 . -4 4.) LI) (1) (4) 'P4) ('9 (N (N I I I I I 4 .I I I I I I . I 1. 1 C). C) C). C) C) C) $4 . . . C) .. 0 0 C) 0 0 0 .0 0• C) - -4 .- - 1-4 -4 .4) ., '4 4 -. '-4 -4 -4 '4 .-4 -4. -. < .)< )')< ).< '4 4< < x 4< x )< )< . N mt . C). 0) C) . 0 0) C) '.0 LI) 0 ' LI) U) .0 '.0 . C) C) p••) C). 0) '.0 4) . N) C) 0) '0 N) -: (N'.0 ,-; - (N (N U . -4 -4 • . N '.0 '.0 If) LI) v to ... 0 1 . 0 - I I 4 I I I 4 I I I I. -4 0• I 0) H C) . 0 C) C) C) )< -4 • C) 4.) : . -4 . -4 1-4 -4 . - 0-4 -40 . ,4. ,4 L/) )< U) 4 0 $4 .. )< )'< )< x •• )< •X < .< < <(N .L/)N >4 cd co 4/) - U) . -4 . U) - LI) -4 If) -4 . L/) _ .. • • . U) . . . . (N ('I). 224 - - C4 t' . . .. I •I• I I I ,'I• I LU LU LU LU LU LU • ,. Ln • • C'I LI r. 0 :• C Ira Q • CN Lh . t-• t' IM 00 • • • • . • • . • 00 0 At . • • • • • • •. • • •• • • . • • • Ira • t') • • •• • • • V) O. 00 • • • • • • • • I I. I I. I I. • • •• • • • L1.4 LUIU LU . • • • H Am Lf) 00 Q . • • - . 00 ' • • • • 0. V • • • ... . • • • • • • • . 0. aM • • . .• ...... • . . . 0) . -. • • . I) O. • O c'a • • . • . • •,. . . • . . . .. ' V • • 0. •...' .. . • ...... V . • . . • .. . 0) . . .' . - . . aM. • . •"••: •...... • . .. .. . . .. • . • .... . .. • ...... ... •...... -. . .. .. ... . .. . . • c6 • .• . • .., ' V • • . . . . • . . ..-4 . • ô • • - . • rsi .e . ra N- . • - . . . . • . • .., ,-• I. .1;' I I I I •l.•. • . . . . • . ... ... I LU LU 'LU. LU . LU . LU LU 0 . . . • . . . . . . . . . • C • • • a. • N- C) 00 • 00 aM • 00 ' . 0 aM 0 aM 0a 00 0 0' II : • • . . • •• LI) •O - C4 i1 . . .. • • . CD .. . . • C4 4 C' •. 00, aM' . •. • . • . . . . V. • ' :: : : : : 226 it was found that a field length of 4 is adequate in the first phase of the program. By variation of meshsize at constant field length (4) the accuracy of the procedure was tested at very small times (T=10-3 ). Pour- point extrapolation gave the derivative predicted by (21) to 8 decimals. The number of meshpoints (401) used in further calculations limits the accuracy of the results to 5 decimals. 6. The results are given in Table B.1 and in Figures 8.1 - B.4. Part A of Table B.l shows for comparison the concentration gradient predicted by the short-time approximation (22). The pure-diffusion approximation (1/Vrr) holds, to within 1% accuracy, if T < 0.04. Addition of the term in (ac/an) = l// + 3t/4 (8.36) extends the validity (within 1%) to T = 0.43. The third term in the perturbation expansion does not improve the approximation since it has also.a positivesign. (See Figures B.! and B.2) At T = 0.43 the flux is 5% higher than the steady-state flux. Part B of Table B.1 shows, instead of the short-time approximations, the value of (30 2/an) , i.e., the deviation from the steady-state flux. This quantity is also plotted in Figure B.•4, as a semi-logarithmic plot vs. time. It is seen that for large T.(e.g., T >0.43) the flux deviates from the steady-state value as a negative exponential function of time. By correlation of the results of part B, between T = 0.85 and T = 1, one obtains log - 1.1198465 } = 0.0522 - 3.134 T (B.37a) '4 0 • • • • ( 228 or f - 1.1198465 = 1.1276 e 76 T (B.37b) At t .= 0.428 this expression gives (c/) 0 = 1.17123, i.e., within .05% the exact value (1.17166). The negative exponential function of time is to be expected since, from (5), the concentration appears to be separable: c = c() + A(t)C(-O) (B.38) where c, is the steady-state function. For C(n) the equation to be solved is: 2 • + 3fl2 dC + A2C •= 0 . (B.39) d • At sufficiently long times only the lowest eigenvalue ofthe Continuous • set is important. This is: = 7.216 (B.40) according to (37b). • 7. The results can be compared with the numerical solution obtained by Olander for Sc =.10 3 .Olander 144 gives values only up to 4t =5, i.e., T = 0.1535. They are slightly lower than the present ones (see Table B.2), in agreement with the trend of Olander's results for increasing Sc number. An approximate analytic solution of (5) has been given by Filinovskii and Kir'yanov, 146 by Laplace-transformation of (5) and 229 Table S. 2 Gradient at the disk interface at Sc = (reference 144) time (ac/sri) 144 . ct (Sc 10 ) T . Olander. •. present . work 1 . 0.0307 . 3.23 3.243 . 2 00614 224 2322 3.. . . . . 0.0921 . . 1.87 •. . . . 1.928. 4 01228 166 1702 5 01535 1515 1555 • • • • • • :.The Note: values in col. 3 are read fromFigure 2 of Ref. 144. • . • • • (Nu ss at Sc =10 is 1 08744) 230 introduction of the auxiliary function i(s) = i(s) exp(-T13) This yields dn If the term in is neglected, (42) can be solved Finally one obtains CO e3CnT = ______+ , (B 43) where c n are the zeros Of the Airy function. In the limit of short times this approximation leads to 5/2 () = + - ___ + O(T no ' ) (B.44) V1TFT 4/Ti which is different from (22) only in the third term Equation (44) is accurate within 1% up to T = 0.667. In the limit of long times (43) gives (fn = 1 0514127 + 0 616845 e4 86346t (B 45) This shows that the neglect of the term in . rj in (42) restricts the validity of the approximation to short tiisi. In figure B.4:it is also * The first seven zeros are given in the second statement (DATA B!) of subroutine APPROX, see eüd of Appendix. Above n = 6 the asymptotic representation c = ((4n-1)) 2" 3 is valid to S significant figures 0 •c.) • • 2 232 • seen that the ti'ansient . ctually. decays faster than in the approximation according to Filinovskii and Kir'yanov. . Finally it is of interest to compare the actual solution with one • based on pure diffusion in a restricted domain, namely in the Nernst layer, of thickness . . . = F(4/3)(v/w) 2 (/3D)a 3 , . (B.46) i.e., based on the steady-state solution (10). According to Cottrell 147. the solution is, in terms of the variables (4):. • ac + 2 e2 377 n2T} (B 47) = r(/3)4 n = ; Figure B.3 shows that this solution underestimates the flux considerably • in the region 0.2 < T < O.S. At long times it approaches, of course., to the exact steady- .state value, according to . ac T = 1.1198465 + e 2.3771 • ( B.48) Figure B.4 shows that this is considerably faster than the actual transient (37).. 234 C1NST2(tN°;JT,!jT'JT) DT4S1J ('fl ,XI(Ul ),E(601 Z(4)1),CC(4i1,.C)(4)1) 11. F' A T(14,,..3.A) 1 POR f(l)(X,Fl).3i ) 114 )'i4T( '1',F1 iic Fi•.T( 1H' 116 F A T I )X 9 v4*4j= 9 1494-1 4= 9F.4941 )T.F3.6) j17 P (T( I 111 F3'\T(l-iD) 113 F0.T(1fX,1):,(y(LF 9 14,6149 .)r= .F3.6) 114 "AT( X,T6,F15.8,715.) 115 F'1AT(41X,1,I5.9) 116 'AAT( 13) 13 RA) 1)1, NJ, -$ F (J..0.1) ST1 C READ ''J4.E r 14 (:Y(LE59 ST I4LTIAL I IiE L:C.r4T .'EA) 1169 K C l' = K ( I N —7 °9IMT 105 P'-IASE '4Jl='4J-1 JL4G=0 C I4ITIAL CONC PR)FtLr.S/ C(J) = ..) / CC(J) 4IAL PR0FIL 0') 1 J=194J Z( J)='-l*FL)AT( i—i) I C( J)r) • 'Rr'JT 10 4ioNJ,-4,')T 'Y) 1() .119N.J KT=r LL=' 5 =1 5 KC=l C 5TAT A TIlECYLE ( 131 OOINTS9 EXCE?T FIST )ME 201) KT= 12./H/ 4 — ) T/1 ) ./ 4 *7(J)(l.+ 7 (J)Tj,.) G=C(J) - T + G'). T') :n 17 A=1.51/H/r7(J) = l./)1 +2./-/H/3./r 2') XT(J=('—xT ((Ni)=1. 0') 4 1J1,N.Jfl J=Ni—Ji 4((J)=XHJ)+r(J)C(J4.l) IF (KT.N.?51 3 ,j T) 4 C STORE C ?flFTLE P5j.3 T4Ifl')•'1 TI '1SE 235 )1 1 J1,Ni C(j 34 C')Z' +3. 4 T2.*.1/Tfl.1415;2,4) PRINt Ill r 2R14y )JT"tT 4T F'JF 1?y T!1-STEP P.f4T t4.LL,T, • S1RT-i" Axr.rr-4 3.141 592654) r1 r)TTFtJ coL , r1T4 r 1. )0 Tc1r9= 4'1X1'.'Ar1') ' FILI'V)VSKIT CALL A ).DrX(T.TSJ . 1) 2l'!t T\tT 1,C'1,T, i IF f JFLA5.i.1) ') Ti 21 =(r)C) +.123i7;17)/r,5.5 XC =)(7r)Y -1.11?R4A ri 2 1 FX , .. iCr)Y =FX( +1,119346 C LG - I0-TT1E S FL . T fI_I..LT • ) T') 3 49 1.)4,T,)c)y 9FXr ,FLG TSJ'!=Tc' 1-1 g . Orr' 1 i, - 1.rs:lM IF (T.'T..;)) ( -s.-1 1(17 IF ( DT=fl1-l. PRINT I )((j) 'J19'J) C • CALJL-.T ST i?-ST.T OFILi .-3y .IITFRTID,' n - ) C F3' 'E CJC ?O&ILE 3Y .J3TACTI'3 ST)Y-.'TAl ?j_ILr: 33 .J=tN.J - RINT !1 . • ORT.\T <1=75 JLA'1 () T'. , 7 E) • rTV)' '(X) C FRF r/ T('II1RAL •)• i 1(1 X. C JStMG AN A?X)'T!N 1) A3rI?4s. A33)LJTE F RR3 ,i 3-7 )IiST')•N A(s) )AT\ 19,i7'c2 v)78'+ / T(X.GT.2.3S)'Y) T)1 r)') III 1296 1') T=(T+(1))*X F 1=1.1(1+1.) çr- **1 RTJPM j ERFC=rXP(_XiX)/X/( .14 1 S 6 94)*O.5*( 1.1-1.5/( j.JX) F 1 F RTJR.N SI JTiN 17 PAPDRI)X (T,TSIft) )T'MStiN C3 (7),A(7) OATA 3/2.3 35 1 741'.7+'4495.5203- 1d3,,. 76(3 138U97. )•#13 33'. 19. 1,1.1 1 17434/ r)r ii •j=,7 U SIIl=S+A( j) 7) 1.5 • 'T'IRN • • V) +4114-.01 237 Appendix C. Degree of dissociation of HSO 4 ion in the system CuSO 4 - H2504 -H20 The dissociation of H 2SO4 in solutions of 0.1 M to 3 M concentration isvirtually complete in the first stage,toHS0 (K 103 mole/i). In very dilute solutions the second ionization constant KB a a H S0 HSO4 is 0.0104 mole/i at 25 °C. Except at infinite dilution the bisulfateions will not be dissociated completely. The stoichiometric dissociation constant KBI c c _/c - , (C.2) H S0 HSO4 varies by three orders of magnitude as the H2SO4 concentration increases from 0 to3 M. The values of Ki which follow from spectroscopic investiga- tions of aqueous H2SO4 solutions,:lSl have been correlated 125 with the true ionic strength by the expression 5.29/I Zn(K0.0l04)'/ = r. (C.3) B 1+0.56/I r Equations (2) and (3) can be solved together with the equation of electroneutrality. Figure B 1 shows the true ionic strength, 'r' and figure B 2 the degree of dissociation, a, of HSO ions in aqueous H 2SO4 solutions, both as a function of H 2SO4molarity The degree of dissociation of HS0 goes through a minimum (0.268) at 0.4 M, then increases slowly, it is 0.33 at 3 M. The ionic strength in the range of concentrations 239 0.5-3 M is represented fairly accurately by r = 1.6 c . (C.5) The addition of CuSO4 influences the bisulfate-sulfate equilibrium by the increase of ionic strength in general, and by possible inter-ionic effects between Cu 2 and SO or HSO ions. Neglecting the latter in • first approximation, let us assume.that CuSO 4 is completely dissociated and that the increase in ionic strength affects the bisulfate equilibriuni in the same manner as if the solution were H2SO4 -H20 only, i e , according to (3) To find the composition of the solution one needs to solve a set of 4 equations (2), (3), electroneutrality and a material balance for total sulfate The equation to be solved, in terms of known H 2SO4 • concentration (4), CuSO4 concentration (C), and unknown HSO ion concentra- tion (x), is: (A+C-x)(2A-c). 5.29V4CZx 1+0.56'SA+4C-2x The degree of dissociation is best expressed in terms of I/K, i.e., the stoichiometric ionic strength (based on complete dissociation of HSO ion), divided by the stoichiometric dissociation constant Figure 3.10 gives the results Note that for 0 M CuSO 4 I/K' - Bct(l+a) C 7 i e , a maximum in I/K corresponds to a minimum in ct(cf Fig B 2) Table C 1 gives the true ionic strength, 1r for various compositions 04 0.3 241 Table Cl. True ionic strongth (in mole/i) of CuSO4 -H2SO4 solutions at • 25 °C. (The values between parentheses are based on partial dissociation of CuSO 4 according to eq. 9). M Cu504 + 0 0.01 0.05 0.10 0.30 0.60 M H2304 + 0 0.040 0.200 0.400 1.200 2.400 (0.0281) (0.118) (0.222) (0.678) (1.153) 001 0.021 0.058 0213 0412 1.212 2.413 (0.051 (0.175) (0.335) (0.978) (1.916) 0.05 0.087 0.121 0.269 0.464 1.263 2.467 (0.115). (0.235) (0.392) (1 033) (1.972) 0.10 0.162 0.196 0.341 0.523 1.329 2.536 (0 191) (0.310) (0.465) (1 104) (2.043) 0.25 0.386 0.421 0.564 0.752 1.542 2.752 (0.416) (0 536) (0.691) (1.325) (2.264) 0.50 0.770 0.805 0.950 1.138 1.922 3 131 (0.799). (0 923) (1.078) (1.710) (2.645) 1.0 1.569 1 606 1.757 1.948 2.731 3.934 (1.601) (1.726) (1.894) (2 513) (3.440) 1 5 2.403 2.439 2.593 2.786 3.571 4 770 (2.433) (2.559) (2 716) (3.341) (4.258) 2.0 3.252 3.292 3.446 3.641 4.427 5 622 (3.284) (3.410) (3.566) (4.185) (5.090) 2.5 4.117 4.156 4.312 4.507 5.293 6.483 (4,148) (4.273) (4.427) (5 038) (5.931) 242 The program INCOP, which finds.x from (6) by Newton-Raphson iteration, is reproduced at the end of this appendix. Strictly speaking it is also necessary to take into consideration that Cu2 and SO ions show a strong tendency to association. This is characteristic of bi-bivalent ionic salts (cf. ZnSO 4 , MgSO4 ). Spectro photometric methods have recently been used to determine .a dissociation constant in dilute solutions. Earlier methods used conductivity data and cryoscopic measurements. The association effect can formally be treated as a dissociation, similar to that of HSQ. Spectroscopic data for the whole range of CuSO4 concentrations solutions are not available, but by the use of semi- empirical expressions the "degree of dissociation" has been calculated from conductivity data and molar activity coefficients. FigUre C.3 shows the values of 0, the degree of dissociation, given by Fedoroff) 48 From these values the stoicbiometrjc dissociation constant can be calculated =c C Ic - - 0 2 c/(l-0) (C.8) Cu 2+ So4 HSO4 and correlated with the true ionic strength. The expression arrived at isof the same form as (3): 15.OSvT r £i(K'/K ) = , (C.9) C• C 1+1.8244 r where is the constant at infinite dilution, 0.00495. Figure C.4 shows the correlation (9). Equation 9 can now be used to assess the importance of incomplete dissociation of CuSO4 for the ionic strength and for the parameter I/K. 245 TâbeC.l contains in parentheses the values of 'r based on (9). Differ- ences of >10% are found only for the solutions most concentrated in CuSO 4 and least in H2SO4 . Figure 3.10, therefore, shows the effect of incomplete dissociation on I/K only for the caseof0.30 M CuSO 4 and variable H2SO4 concentration. The effect is to decrease dissociation of HSO; this implies that the decrease in ionic strength, and consequent increase in the activity coefficients of H and SO ions, outweighs the effect that the decrease in S0 ion concentration has on the dissociation equilib- riuin. Finally, Figure C.5 shows the analogue of I/K 1 for the CuSO4 dissociation (or association) equilibrium. is an index Of the degree of dissociation of CuSO4 in acidified solution: I/K' 1(1-0) . C C - (C. 10) so4 The addition of 0.5 to 3.0 M H2SO4 causes a fairly constant degree of dissociation, 0 08, independent of the CuSO 4 concentration. The program INCOP (see below) takes the association of Cu2 and SO into account after first calculating the ionic strength as if CuSO 4 is completely dissociated (from equation 6) Then the bisulfate concentra- tion is treated as a constant and the CuSO 4 dissociation is solved for. If y is the concentration of Cu2 ions, the following equation is solved by Newton-Raphson root-finding iterations Lii y(A+y-x) - 15 05V'3A+4y-2x (C-y)0 0 (C 11) 1+0 56v'3A+4y-2x - '2 248 PROGRA9 CI°P(INPUT,OUTPUT) C CALCULATES IISULFATE cONC:'1rArIuN (X) AS A FJNCTIUN UF C iP PER SJLFAT C COMCE4TATI)M (CC) AND SULFURIC ACID CUNCENTATION (A), J3ING E).(7) U C UCRL-19102 AN) NEST RA0 15OM tODT. FIN)ING METHOD. C REpEArS CALJLATION rA 16 F)MAr(r5912F9.6,2I.) 18 FIf1Ar(1lc9.4) ') FIAT(?I) 28 FOR'4T(1H3) REA') 199KTE'1,LTERN RA' 189(A(J)9J19LTFr'1) REAr) 18,(R(J),J=19KTFRM) PRINT 4 PRINT 8 KK=) ro 21. 1=19'(TFR4 IY) 71 .J=lLTRM PRINT 28 DEv = . XIL=1. $ LL1 • .AAA(J) CC=RU) RR=A4/(CC+A) AAl=lA S CC1=CC C JF= ASSO'!AT ION ('15)4 NOT TA • v=r- I !F(JF,I.fl Y=).5CC C ITFRTE TI FINt) X FIR rixEr) y 1, r <=1,2') XIA=3.4A +6.'C' • IF (X.G.(2.'At l.?41IAA IF XI = l.*4A 4V .)•*'< F=FF+ALr).((A+y 0FDX=(X**2 —.*AA* —2. 4 AA*y )/(4\.Y — X)/(2.wA—X)/X 1+5. 1 /S)'T(XI )/( 1.+).SEc*c)TIx )2 XXX - F/1F1x I (XX.LF.').) XY=i.)1)111AA IF (ASXX—X).LT..1I011) ') TO 2 I X=XX. PRT'T 7 ' X=XX C ION CJCE4TATI3N Cl=X C?4A+Y - X • XK lC1/ AAA=(C?—Y)/A.! & • CrC=Y/CC X!K=XI4/X< 1 ( Jc N $~ . 1 10 T) TF (''.'.).) ,.,t) To 21 C ITrAT TI FTNr, v POR FIX. x c)r) 11 L1,O GG6+ALO'3 (Y*(A+Y_X)/(C(:Y)) — 13.)5*SQT(XI)/(l.s.1.:24*SRT(xI) )GDY=_((Y*2 — 2.CCY +('X —A'CC)/Y/(CC—Y)/(A,*Y—X)) yv=V—.r/r)GIy TF(YY.L.1.YY=.).))1rr tFV.G.'C) ''Y.9999C( rF(A1cryy_ , ).LT...1iI.1, C) TI 9 Ir) YYY PRINT 7 9 YVV SC4 = YS C2:AA+YX A4A=V9—Y/AA S 1k=C1/A4 t XKK=Y*C2/(CC_Y) XIKK=XIA/XKK IF(S(XTxrL).LT..1OOr1) GOTO 11 XTL=XI S L.=LL+1 TF(LL.LT.1) ('1 TI 12 PRINT 7 ii ?RNT169K , RR,AA,C(,XIA,XI,XK,XIK '-'AA933,CCC,XK9XI +I 1 .1'"11 1 I5+I'.II1I+"1I+.I11+11. 1+11.1I1..'1.25'1+11. 1 .1r-e-11 250 AppondixD Migration at a rotating disk electrode in*idified solutions of CuSO4 characterized by the bisulfate dissociation parameter The numerical solution follows the lines indicated by Hsueh and Newman' 25 The equations of convective diffusion for the H', HSO,44 S0 ions now contain a production term involving the parameter I/X. (all concentrations being made dimensionless with respect to I). The program MIGR is reproduced at the end of the appendix. The migration effect on the current is shown in Figures 3.11 and D.l. The results are collected in Table D.l. Note that the solution pre-supposes (1) constant physical properties, (2) no association of Cu2' and SO ions. In connectioi with (2) it is interesting to note that Harned and Hudson, 149 testing the theory of the electrophoretic effect in diffusion of bi-bivalent electrolytes (ZflSO 4 , • MgSO4 ), obtainedgood agreement withthe theory if a separate mobility was assumed for the ion-pair, approximately 40% higher than that of the • dissociated electrolyte. This was explained by loss of water of hydra- tionof the cation. Since migration effects depend on the mobility ratio of reacting ion and supporting ion, CuSO4 association could account for a higher migration contribution to the current than predicted by • computations based on complete dissociation of CuSO 4 and nearly-complete dissociation (I/K5) of HSO4 ions. 251 Table Dl. Migration current and electrolyte composition at the electrode A) i L A * H + 0.020 0.015 0.010 0.005 0.0025 0 r. 0.99 1.124897 1.124886 1.124878 1.124873 1.124930 1.124872 0.95 1.145449 1.145437 1 145428 1.145423 1.145599 1.145421 0.90 1.172204 1 172189 1.172179 1.172172 1.172495 1.172170 0 85 1.200227 1 200207 1.200193 1.200194 1.200682 1.200181 0 80 1 229608 1.229581 1.229561 1 229550 1.230232 1.229546 0 70 1.292807 1.292752 1.292714 1.292690 1.293731 1.292682 0.60 1.362492 1.362380 1.362300 1.362251 1.363569 1.362235 0 50 1.439323 1.439087 1.438919 1 438818 1.440262 1.438784 B) 1SO4 CuSo4 r + 0.99 0.297586 0 297584 0.297583 0.297582 0.293610 0 29759 0.95 0.289009 0.289008 0.289007 0.289006 0.284910 0.28901 0.90 0.277906 0.277906 0.277907 0.277907 0.273696 0.27791 0 85 0.266385 0.266389 0.266392 0.266394 0 262122 0.26639 0.80 0.254460 0.254471 0.254479 0.254483 0.250211 0.25549 0.70 0.229508 0.229544 0 229570 0.229585 6.225568 0.22959 0.60 0 203365 0 203453 0 203517 0.203555 0.199926 0.20357 0 50 0 176456 0 176659 0 176798 0 176881 0.173900 0.17691 by three-point extrapolation from H=0 005, 0 010, 0.020 . Number of meshpoints 802 Mobility ratios used H/Cu2 = 12 95, HS0/Cu2 = 1.852, SO/Cu2 = 1.482. U. 0. 253 '21) (13A1 MI OR> I NLJT .'1 I IP IT ) C 2200RAI FOR FFFECT OF J ATJO ON L IN! TING C):FNT C INlPLFr orsslCIAT ION IMCNS ION 1) 5,5) 3L5 ,5 ,ç 5 02) 5) (5,5) ,Y( 5,5 ,j lv) 807) '11 I 1 5) )9 71 5) ) ' I 1) 5) I Cc)M'0N ill 01?'AAT I314,RP.4) 112 FORA4T (48.',AS) 103,FORMAT (4HO 14J:914,51, H= , F6..o/34SpFCrF S LI OrE 7 1 S/(IX,46,7F8.5,7r5.1), 114 rORMAT (I2H1NFRNST STAON4Nt DIFFJS ION LAYER> 115 FORMAT (32Hp320,JING OROP ON PLANE LFCTR:Jr)F) 116 1RAAT (14H191TATT'io 11S1) TI' FOR"i'f )5)3.4) .118 FORMAT (3IHTTHF NCXT Ri 110 NOT C1NVROE) 109 FORMAT (lH),X94F10.6/)3X,697F9.5)) lie) ORMAT (35HISPF(7IF 4; (INF CTERO AND) )9 REAl 101, HODE,NJ,N,H 6 IF(HOiOC.EJ.J) STOP 5(_3010 (1,2,3),5Q0E 1 H= 1.1/IN)-?) C')NST=l.) %AMDI=l.J 5 3 RINT 114 500 TI A 2 CONST=).) A"PDl. 12J . SPRINT 105 500 10 4 I CONcTl.0 SAMP0r1.1)q 6?RINT 116 4 lMl=-i N'1=42 S CR1'.O REAl 102, (0) 1 ,OIF( 11,7(1) S( I) 92E( I I PR I NT 1 J 3, NJ H, ) N F I) ,j( )) ,O1 FILl ,Z (I) S(I)9 1 1 q N Al R I N T ill 5 REAl 117,XI,R.AK TF)XIEQ.0.o) 00 10 99 £ •1N(N).1 cA=xI*(1._p) , )4._R) S.CR=CA*R/(1._R) CIN(4)=CA S IF(C3+\K.OT.CA) GO TORTS CIN ( 2)=0.5*(CA_c3_SK+R1) $ GO TO)? 3 32 IF(CA+AK.OT.C8) 00 TO 3.3 .6 CtN30.5*c3_CA_AK1( GO TO 34 33 CJN(3):4.CI*AK,(Rl ~ C4+AK_C3) 14 CIN(1)=C.IN(2)*1N(),,\K 6 1)0 7 J=l,NJ V(J)=' CONT*IH*( NJJl I (MOOET) IF) NMT) CR1 + (cTN(NMI) -CRJ)*NJ_J_iI/I.NJ_2l C(N,J)= 1O 10 7 119NM7 7 CIT 'J> = CINI I S JElINTrI S 8 JCOINT= JCOIJNT 1 5 AMPI5 AM £ J=O 00 9 1=1N 6 00 9 K=1,N S 9 X(I,K)= 1.0 11 .3= ,j +.1 . DO ii lri,N o 5 D ii K=i,N S A(1,K(=1.i Ii D( I9<)=0 • £ IF(J.GT.1) :00 TO 14 5 10 11 1z19N 6 3) 1,1 )1. 13 G(T)=CTN(I) S CALL 3ANO(j) 00.10 .10 14 Do 1 5 . K=l 9NM1 . 15 '3(N1,K)=Z(K) S IF(J.PQ.J> 00 TO 18 5 .00 17 I=l,NAT PP= i( )/OIF.( I) *(C(N,J+i)_C)N:,J_.1 PP= 1(1 (/OIF( I) *)()N,Jfl)+((N,J_l )-? 0C(\,J) I CIT 'i - I)) / .1 4) I , Ti = - 1 • 1 + / ' • 0 - H V C.) I /2 • I / I) I F>1 POP 1*2(J) /2. 1/01 F ( . 1 \) I .N) 01 1) /11 F>! ) (CP/2 • 0 - ) I , N) 2 • 1*1(T) = /01 IM IT , I I 11 T,N)= - 31:! )/ITF) I) 1c/.o C) I ,J) I . .(%)j) .7 - po*(,J) 254 DO 23 1 = 23 I A( 1 9 1 1 = A)l, 1) 5 3(1, 1) r3( 1, 1) 13)1 , 1) (1,1) A (I 'N) =A( I 'NN fAll 'N 5)1 ''fl =8) I'N ) +0(1 N) S O(T 'N) r)) I 'N) *13)1 'N) 23 o i r I I + (3 (.1 ) :6 All 9 H = s .1. 5 0 (1 9 1) =AK s .o (1 v 1 =3 • S All N ) = 0.:) 5 31 1 9 N ) =0. i S )( 1. 'N 1=2.0 5 31 1 ) =C (3,0) C 1 2 ,J 3( 1 •3) Cl 2' J) 5 0) 19 2) =C 1 3 9 J) 5 C ALL lAND I J :5 : TO 10 18 130 19 T=2,NM7 0P il I iori Vi T 9NM 1 )= S) T )*OIF(NM1 /c) NMI /131 2 )1) Yll , I (l .0 5 Al I, I) .PP 5 3(1,1) =1 .3 5 Ri I ,NAl ) V (I ,N) (S( 1) IS) NM) 1 *J) NM1 ) ()•) 1) C (I ,NJ- 11)/1)1 F) I I • 15(1 ,N)= - V(I ,N) 19 DII :PP(( 1,NJ-1 ( 5 G(NOi)=CRO 5 A)N0l,NA1)=1.D b 13(1,11=-AK • (3(1) = C I I ,J) C ( 2 J) 5 0(1 '3) =CI 2 ,J i( 1,2) C ( I ,J) CALL 3ND) ) A"IP= (UINA 1) .*Cu*(c(N , NJ_n_C(,:4J, ( - D(F(Ni'j1);(C()I9J - 2)c()I'i19 H 1114Jn/..5/HICIN(NM1 NMH • DF.L2.(C(i,NJ-1)+C(3,N.J-1)-CIN(1)-C1N())/CT\))4) kAM°=AMP/AHP) S 3) LI 22 T(J(7JUNT.LF.1:)) (31) TO S I P4TN.T 118 22 PtR I NT 1599 AMP,RA)IP , AK,1)FL, I 2FF) I) , C iI9l)qc(Iv ,'4J-11 , I 1 'r TO s S;J')UTINF 134N13(J) • I) tMN TON A) 5'5) '3) 5'5) 'C1 5,8:32 ) 'I)) 5 '11) '(S(5 ) Xl 55 ) V 55 1 ) S • 6' 8 Il 7 ) • (fl'M0N A,3,(,O,(3,X.Y,N,N) ill PORMAT .)l511 ) 17 rFR 11=3 Al 1=,14) TF(J.i1.l) DO TO 6 5 N?)= 11+1 5 50 7 ibN I 313 7 L=l.N 6 LPN=L*' • 2 DII ,LPN)X( T,L) '5 (.AL.. .1AT1NVLN,2)N+1,DlF%'1) 1FI13FT'.0.5.0) PRTNT 1)1, ' 1) 15 5 K=1,N (K,NPi,1(D(K,2N+1) '55)3 Ll,N 5 F(K,L,1H-D),L) LPN=L+0 5 X)K.!)=_15.(K,LDN) S 6 IF(J.'ST.?) 00 1') 8 5 DO 7 I=l,'J 6 30 7 K=j,N 5 131) 1 L=1,' 7 3) 1 ,K (=131 Ii K' I I .L 1 "XI L 9 K) 8 TF(J.LT.NJ) (3010 11 15 130 10 11.N & 133 13 L=i,N 0111=3)1) - VI 1,L)"F)L,N1,J-2) 5 133 I v=i.,N Al I eL) A( I ,L) + V (I ,M) *F( H 9 L,02.) 11 00 12 11N 16 DII,NPl)=-0( I) 5 50 12 L1,N' H 011 eN"l ) 0( 1,NP1 ) A IT eL I' L 9 NP1 9J - 11 50 17 K 1 , N 12 9(1 eK ) =31 1 ,K I •+ All 9 L ( 'F( L 9 K,J - i I L CALL ;AT I NV) N 9 NP1 , IFI3ETI41.:O.3.s I POINT ill,J S ND iS .K=1,N 5 13:) 13 1,1z1 ,N1 1.5 FIK,M,J)DK,)'1) $ !F(J.LT.'J) OCION b 13) 17 K=l,N • 17 ()K,J)=F(K,NPI,j( $ 13) 18 JJ=29NJ S 1=00-00+1 .b 5:J 10 =19O. .C((,'i)=E(K,Nl,1) 5 1313 18 L=1,N • 1.8 C(K.M)=C(K,'.l) + (K,L,'1)C(L,H+1) 5 00 13 Ls1,N 5 130 1') K=1,N 19 C(K,1 )=C(Kl), + XIK,Lc(3,L S 2TJkN 6 FND H • SUOROJITNE 'IArTN\I( N,M,0TF2'4) S C3'400N •3.3,C) C :I9fl( I NVEOSION W I T H ACCu1PA'Y 103 SOLJTIO.N JF L Io5,1 E 0AT rols.' 01 0N5 TON 51 '3,5) ' 3 ( 5,5) ,) Se 832) a!)) 59111 , Ill 5) 1)FTR'11.0 5 1315 1 H1,N 1 113)1 )=1 3 4 130 18 NN=1,N I R 1,'AX=3.i $ 50 6 T=l,N IF( 113(1) •NE.1) (313 TI) 6 1 00,5 J=l,N 5 IF) 1010) •NL.) ) N) I:) 'S I'F(35('3(I,J)).LT.3MAX) (3:) TO 5 5 0.iAXA8I(3( 1,.))) 5 I'OY'JiSJCJL=J • 5 CONTINUE 6 (T0NTTN) 914 7 13FT'1') S . RT)RN 8 113(JCOL)z'1 6 IF(JCOL.E0.1ilDW) 0') TO 12 5 130 13 J1'N SAV0(I0N,J) 5 3lI50', ))=R(JCSL,J) • 1.0 :3(JCDL,J)=SAVF 5 00 1i K=l,:'1 S OAVE=u( IROi,K) L( IO0.I9.KI=5(JCOL,.<) • • : • . • • : • • • 255 l 13 l (J L,J)=RLjr)l,J)rF .1; ') .14 K=1,i 14 D(.JOL,)=(.JCO,K) F I J 18 IzI,N IF( T P- J)L 3) f 18 • • • • • • P8(T':)COL) $ D 16 J=I'N • • 16 R( J 9 J)ri1( I - .F*1(JCQL,j) •&DO 17 • • • • 5 175-7 • • 8SO4- 1661—.1-7i— 49R—n 41 —i I 1''—) • . H+ 7 1 4-1 7 9—O +188 1 )+88)_1+78808_.1 . +1 8 8+fl+S511')_+2I. -y)_ I . •• 4-188884-9+A8nrl-1 Appendix E. Theoretical solutions for transient convective diffusion. The values of transition times in Tables 3.4 and 4.1 are based on the:following relations. 1.1 Free convection at a vertical plate. Exact, numerical, solutions for a step in concentration at the plate have not been found in the literature. Exact solutions for a step in the flux at the plate have been reported in the heat transfer literature. These will be discussed first. Gebhart 157 used an integral method to compute solutions for various Pr numbers. He found that there is a rapid initial approach to the steady-state temperature during which the pure-conduction transient is closely approximated until about TI T = 0.75. The remaining part ss of the transition is slow and shows a slight (- 1%) overshoot for Small Pr numbers. Expressed in convective diffusion parameters and variables: * 4 Sc; Gr. = (gxL N)/(v 2D); and *Dt * 2/5 - .-. (Gr Sc) , (E.l) L the transition times are: = 0.90 0.99 * Sc = 0.72 T =1.73 2.73 5 1.36 2.34 1000 1.12 2.13 It is of interest to compare the results for Sc = 1000 with transi- tion times reported by Tvarusko and Watkins 68 from interferometric measure- ments at a vertical cathode in 0.23 H CuSO 4 solution. The transition 257 times for various current densities as indicated by Fig 4 of Ref. 68 are * 1(ma/cm 2 ) t(sec) T Gr ScxlO +10 060 80 444 063 3.92 36 4.24 4.14 •23.5 12.0 2.89 24.8 36.9 9.5 2.74 39.0 * -6 2 In the calculation of T , L is taken to be 0.9 cm, D = 5.4 x 10 cm /see, a = 0.15 i/mole, v = 1.03 cs, t = 0.366. * Cu The experimental T values are higher than expected, in particular for lcwcurrent densities, i.e., small driving force. The transition time following a concentration step at a vertical plate can be estimated from an approximate solutiongiven by Siegel 145 and supplemented by Goldstein. The end of the one-dimensional conductive period (t 1 ) and the end of the asymptotic approach to steady state Cr2 ) are obtained by employing the method of characteristics and assuming similarity of the tranient profiles. The results are flux step concentration step period 1 = 1.71 (1+2Sc)2" = 2 37(l+Sc)1"2 1/2 period 2 = 4 33(1+1 68Sc T 5 72(1+0 S9Sc ) Dt 112 T* Dt * 215 Here T = -. ( GrSc) and = -- (Gr Sc) (E,2) L The results for a flux step are given for coniparison with Gebhart's solution The exact transition time (to 1% from steady-state) is under- estimated by t 1 and overestimated by T 2'In the calculations of Table 4 1 the transition time for a concentration step was taken to be T = 3 It is interesting to compare this with experimental observations from the electrochemical literature, in caseswhere a constant potential was applied. 48 Bohni, Ibi and Frei repor.t a transition time of 3 minutesto reach steady-state in an 0.1 M CuSO 4 acidified solution The vertical cathode was 2 cm high; the diaphragm used in their experiments was close to the electrode but remote enough (> 0.3 mm) to obtain limiting current valuesin agreement with the classical relation for free convection at a vertical plate Using the physical properties p = 1.27 cp, p = 1.10 g/ml, D = 5 3xl0 6 cm2/sec, one obtains T = 10 5 (Ra = 0 192 x 1010) This is considerably longer than expected. For ferricyanide reduction in equimolar solution the •transition time was longer; up to 50 minutes if the distance cathode-diaphragm was made smaller.than the critical distance referred to above (cf. equation 20 of Table 1.3). This indicates that the development of secondary convec- tion, which dominates mass transfer in such narrow spaces open to the bulk •(cf. p.37), may take a much longer time than one would calculate from the solutions.for free convection at a vertical plate. This is also suggested by the observation of Marchiano and ArvIa 66 that development of a steady limiting current at a 1 cm high cathode In 0.005 M CuSO4 (with 1.5.M H2SO4 ), following a potential step to -500 my, required 20 minutes. This corresponds to: = 21 (Gr Sc =:9.75 x 106) 259 again much longer than expected. One should note that the cell employed had a volume of 64 cm3 while the cathode occupied only.a 1 x1 cm area. Moreover, the driving force for the secondary convection was very small. Lvgue-type mass transfer A complete analytic solution for a concentration step at the surface has been given by Solimañ and Chainbr.' 59 Here the :characterjstic time variable is (4 4) I. - . UfA ) 3 t.3) where S = T/p, i e , the local velocity gradient at the wall The one- dimensional diffusion transient is followed to approximately T = 0 80 The approach to the steady-state is exponential and complete to within 10% at t = 0.92, and towithin1%att = 1.25. (Used in Tab1e 4.1) The procedure for solving the unsteady-state following a flux step at the surface has also been described by Soliman and Chambr, but numerical values are not given Rotating disk at high Sc number (see also Appendix B*) The unsteady state at a rotating disk has dimensional similarity to that in Lvêque mass transfer, if it is taken into account that at the rotating disk S(r) = c.3/2v_112r The variables in the two problems are: Note that T is here identical with (4a) of Appendix B, and differs from 2/3 T = wt Sc"3 used in Chapters 3 and 4 by a factor (/3) = 0.307. 260 Lvgue Problem Rotating Disk Dt Dt fcj2D\ 3 T (9Dx/S) 2" 3. (3Dr/S) T3 y - yw112 fv\ 3 (9Dx/S)3 (3Dr/S)1'3 1/2 \) The equations differ by one term only: 2 2c 3c Leveque: ac (1-6rrt) -. 3n = (E.4) • •. , • 3c 23c 2 c • Rot. Disk: - - 3n a • The extra term in the Lévque equation derives from the non-uniform thickness of the diffusion boundary-layer. The steady state is reached more slowly than in a uniform diffusion layer having the same velocity gradient: T = Dt/(9Dx/.S) 2" 3 = 1.25 for the Lvque transient, while the analogous time Dt/(9Dr/S)2A'3 is 0.31 for the rotating disk transient. Apparently the parallel convective, inflow provides a more effective replenishment of the falling concentration in the diffusion layer than the convection perpendicular to the surface. Hale 154 has numerically computed the transient concentration at the disk surface following a step in the 'flux. The transition is 90% complete at r =1.40, 99 % complete at T =3.02 (used in Table3.4). This is significantly slower than the transition following a càncentration step. 261 Appendix F. Transient diffusion following linear potential decrease at a redox electrode. 'I 1. The equations describing pure diffusion of the oxidized species (ox) and the reduced species (red) are: 2 ox_ n ox at Ubox 2 ay 2 red n re. at t red 2 ay with the initial conditions t = Oy > 0, C = Co b , C red= Cred , b (F3) and the boundary conditions ac ac. > o, = , -D ox = D red ox ay red ay (F.4a) y= ,c =c ,c . ox ox,b •edr =c red,b (F.4b) The concentration ratio at the electrode is prescribed by: RT c t >0, E = E ° + ..2_ Pn ° nF = E. + a't (F.4c) • c red• in where E ° is the standard redox potential and Em the initial potential This is equivalent to RT IC /C = £ - E = P.n _OX ( ox,b • in nF I = a't (F 4d) [red1Ic ic. red,b In the following a' is supposed to be negative 262 • 2. In dimensionless variables 0= c /c ox ox ox,b 0red = Cred/Credbj 4) = nF/RT, the boundary conditions (3,4) can be restated t = 0• >' > 0 = 1, 8red = 00X (F 5) ao 'c D >0, y = 0, . ar = ox(oxb (F.6a) Y Y . C redb red ) 0 OX y = 0 4) - Lii = at , (F 6b) red = ' 0ox = °d = (F 6c) lao \ Note that (, o), i.e., at the electrode, is unknown but coupled to O laored \ lao \ x) ("'red) eat + a 0red,o eat (F 7) 3 The solution of (1) for a prescribed variation of c0 at the electrode is, according to the Duhamel (superposition), theorem: = J c0(y,t) (F 8) erfc 2Vt(t-t') (')° dtt ox In dimensionless variables the concentration gradient is then ao ox = do) dt' (F 9) ( 263 For the reduced species an analogous equation can be written. The flux equality (6a) takes the form: \ /dO \ 'ox -R dt j* (F 10) (tt?)h/2 dt' = 112 dt' , where R = ox b ox (F.11) red,bv1 red 4 The integrals in (10) are conveniently evaluated by the method of Acrivos and Chambr. In this method the t axis is divided into n equal dilT) intervals of length Lt' and the derivative ( is written in difference form for each interval t (dO I kAt O(k1t)-0((k-1)t) - (t-t') (k-l)Lt At(k&-t') • (F.12) • • Upon integration and rearrangement one obtains 00(k&)-0((k-1)tt) kLt dt' 1/2 = k=l 3 (kAt-t') (k-l)& (n22 (At)112 O(l) {fl1)1/2 - - 0(n) E 0(k)Ak] , (F 13) where Ak = 2k 12 - (k+l)112 - (k-1) 2 5 Applying this method to (10) one can write 264 • l. ..f . . . 1 . . . . . • (n) (l)B (k)A = •. . . . . RJO ox - 0 ox .n-1 -. .0 ox . n-i I. • •. I . • . k=1 . ...•• . J•• ..... 4.•j 0red - OredBn_1 0redn-k = O0 (n)exp {-(n-l)at} - 0(1)B1 . . . . n-i. . ., . 0(k)expL(k4)aAt}... A , . . . . . .• (F.14).. • ' : . k=1 •:.. - . . . ... where (6b) has been used, and Bk = k '2 - (k-i) 1 " 2 The final expression for is B 1 (R+i) + (R)(s 1 ) + s 2 fl. 0 ( n) •:. . . (F.15) ox . . . . .. R + exp {-(n-1)aM} . where S 1 = 0(k) An_k (F 16a) and s2 00 (k) Afik exp {-(k-1)at} (F 16b) The final expression for the current is 265 /2 cnFoxb (~Dox [B l exP(__l)at) - i} + s 1 exp(-(n-l)a&) - s2 2 F 17 (at)hI2 R + exp(-(n-l)aEt) The program THETRO reproduced at the end of this appendix calculates the current and e for a given value of the parameter R and a given value of the increment aLt, i.e., the dimensionless potential. 401 mesh- points are used in the calculation. The initial increment of aLt = AO is 2.5xlO 5 . In the next time cycle the increment is multiplied by 10 and the first 40 rneshpoints are selected from the last cycle. Seven decades were usually computed in this way. The results for various values of R are shown in Table F.l: the maximum current 1(17) and corresponding potential, as well as at ox the electrode when the current peak occurs, At small values of R the current maximum agrees to 5 significant 'figures with that of (4.28), i.e., for a deposition reaction. At high values of R the current maximum agrees to three significant figures with that computed by Nicholson and Sham, 166 who used a similar integration technique. As shown in Figure 4.17, the current maximum is displaced to more negative potentials as R is increased. The reason is that the depletion of the diffusion layer takes more time. The reactant concentration is high compared to the product concentration and the initial current by itself would produce a larger shift from equilibrium than in the case of 266 equimolar:bulk concentration; because the potential is controlled, the effect will be indirect and the initial current will be depressed. This is clearly shown in Figure 4.17; also, as noted in Chapter 4, the maximum • current will be less. • Reinmuth 171 has proved that the shift of the maximum to more negative potentials is proportional to the increase in in R. In physical terms: at high values of R, i.e., at large anodic initial potentials, • the potential of the maximum will be a fixed amount more negative than 0 RT • E • + in R • ; • • (F.18) if Coxb. = cred,b, (18) is recognized as the half-wave, potential CE 112 ) of polarography. According to Nicholson and Sham 166 the peak potential is 'B = E 112 - 1.109 •' . • • • • • ( nF F.19) Table 1 shows that the agreement with the results of Nicholson. and Shain is quite good at R = 10. At higher values of R the present method' of numerical integrationgives inaccurate results, because the more important contributions to the sums s l and s 2 (16) are made at potentials where the (at) increment is not small anymore (0.025 in the calculations of Table 1; Nicholson. and Shain used a uniform increment. L(at) 0.01. . 267 Table F 1 Current maxima in linear potential decrease at a redox electrode R I(e.F. 17) r 0ox,0 10. 0.7916 8.025 0.247 io2 0.7952 : 5.725 0.249. 101 0.8223 3.450 . 0.265 1.1 0.9346 1.600 0.347 1 . 0.9410 1.575. 0.343 10 1.0566 0.9400 0.413 .102 . 1.0793 0.8650 0.423 10 1 0818 0.8550 0 426 1.0921 0.8540 0.426. 0 (theory) 1.08209 0 85403 0 4257 (Ref. 166) . . 0.7911 + 0.001. :. 7.998 268 PI0 R I T 11rT0( 1P'JT, ) )TIt) N r )N T U 4 ) 1 , 14 ) M, 14 1 ) ,' J ( 4 11 ) ,TAA I 4) 1) 102 FDRMAT(X,19,x,Eli.3,X,E13.(,x,E13.6) 10 FOR9AT I 1HO, )H N?. 10H PHI ,X,13H 1THETA 9 X,l5H CtJRET ) I i4 FORMAT 1)X,4HNLT, I49419 1)L,F12,8,6i,51GN=9.0 9609R-\TE,F12.,15d l,1)ECAr)ES t1i ,16) 106 FORAr(?3x,7.31()x i E11.5,X,F13.6) 107 FORMAT(1lX,'+H) Z 12.5) 115 FOR\IAT (TI) 125 F '4r(115X,-1RATE ,E.5.l) 1. F0R1AT (ii 5X, 13 - V) 7 CA l. , 19 P0AAT( IHI,23X, 7H DjJ ,)X ,1H T I ET4 914H 1CJR1NT ) 343 FoR1Ar ( 1JX,l'+HRDO.X REACT IJN,X5H R 'F12.5) 1.1 11 FOR'IAT( 1H1 ) 1117 FORAT(1H0) C "l\IAX '\J(J OF CASrS ITTH !rFL'HT i REAo 1l5,AiAX 70 T H ( 1) = 1 • 111 FORMAT(14,F12.3,F3,:),E12.5,12,512.5) C \IZT=N J 3E ' TI E-STEP I 3 'h CYCLE , DL= T I M E I' C SI3N=S1N SCAN , R A T E = RAT, N)EC NJAER OF CYCLE.S , R=R RL.Ai) 1119NZT,DZ9SIGN,R4TE,NT)EC9R 1)1=0.5*1)7 .EX0.5 ETA(1)=fl. • C'JR(l)=O. 1EC=0. W4=MM+i IF(M.M ..GT.IMAX) GO TOlill H JBEGIN=2 KKi.AX(.NZT-1)/10 + 1 KK74- (Kfr4AX-1)/11 C F3i1 COFrIcI:'1'r5 OF ACJVJ5-CHAM3. I N T E ,' -i R A T I ~D A P3C)URE 1 )0 61 NZ1,NZT 5 ANZNZ A(NZ)2.3*ANZ*EX - (ANZ+1.)**EX .(ANZ1.)EX 61 3(N7)=tkI? - FX - PRINT. liii PRINT 3439, R PRINT 1)49NZT9, DZ,SIGN,RATE,NDEC • PRINT iOI NK=Nr)fC 1 • DC) I M=1,N< • 1)3 80 NZ=J9GTN9N7T • TW2. • •Z(NZ-l)*r)Z ETA=RATF*SIG1*Z • 0N=IXD1_ETA) Fr)RMJ Si =SUi AND S2=SiM2 S!JM=0. • • • IF (NZ.L.2) GO TO 47 • • • T'.=2. • • • DO 41 J=39N7 • KN7 -.J +1 269 SJ'1=S'i1 + 1H(J-1)3A(K) .-((J--2)RATF*SJC,N*fl7) SfJ1l 5) + TH(J_1)A(<)x 2 (K*RtTESiGNL)Z) 40 SJM2 SUM? +TH(J - l)*A(K):p(P:)) 42 NJ= .NZ - 1 !F(RR.LT.P()N (() 10 77 S • THtNZ ). ( ( NJ) Cl .+R) + 5'J + SU"l? ) / (R EX 3 (-FT A )) GO TO 78 • 77 TH(NZ)=SUl3 7A CJR(NL=T F 1(Nfl - TH\'Z +j/zx. PRINT 1179 117 9 ETA.1i(N7),TuR( Z) I F ( T-1flfl .LE. 1 .0E-2?) ) GD ID ii 7 80 CO'JTI)E 117 00 ii K=1,KK1AX J=K*lo_ 0 T1(K)=TH(J) C!JRIK)=CJRLJ) • ETAA(K)FTAA(J) 11 CONTIN) DZ1:0.*D 7 PR!)1 liii. PRINT 1259RATE PR!Nr l 7 6,If C 2 RINT 107,DZ PRINT 19 PRINT 106,(TANZ ,T(N7) , J iR ,14 Z A Z1,KK1AX) PRINT 111.2 PRINT 103 JBEGT N = KK 'lAX+ I 3 CONTINIE DRINT liii . PRI)T 1112 GOTO7 1) : 111 CONTINUE STOP END +10 +401.+005.0__l.+1.000fl0F_7+1.:'000'rr+05 +/4 0 l+005.01r-_1 .+1.0O000F_07+l,'Vr)+0() +401+005, 000-D5-1, + 1, 000ThE-010 7+1. 0000D-D 1 +40) +0O5.lfl_flS_l, -'-1. 0 ODF-D 7+1, 03 Mq 401+005.')0O-o5-1. +1.00•0.00E-Doo 7+ 1.O00o0E - o 270 Appendix G. Transient diffusion following linear potential decrease in a deposition reaction characterized by slow electrodekinetics. 1 The equation describing pure diffusion in a semi-infinite medium is • - C 2 • • . • G P Dy with the initial condition t = 0, y > 0, C = Cb; (G.2a) and the boundary condition: • • t > 0 )i = C = Cb (C 2b) The. concentration at the electrode is related to the total overpotential, by the expression (2.29), (n_n)ZF ,. i = -nFD(---)DC = 4exp R0T -exp RT . . (G.3) 0 . . . where the ohmic contribution is assumed equal to zero. The concentration-. dependent exchange current density j 0 can be replaced by i o (c o/cb) ' where j is taken at c = Cb• . . . . . The total overpotential is prescribed n=a't, or, (G4a) = flZF/RT at (G 4b) 271 In the following a is assumed to be negative. 2. In terms of dimensionless concentration, 0, and potential (4b), the boundary conditions are: t = 0, y.>O 8 = 1 (G.5a) t > 0, y (G 5b) t > 0, y = 0, By cbnF (GSc) where the Nernst relation (2.22) has been used 3 The solution of (1) for a prescribed variation of c at the electrode, (dc/dt), is according to the superposition theorem t c(y,t) = erfc —k-' (.E--) dt' dt 0 o 2/ TF 5 In dimensionless variables the concentration gradient at the electrode is then: () (dtT) o dt' = f D(t-t') The current condition (3) can now be written (dO/dt') 0 acat 0 Y+ate = - o - dt' 0 0 cbnF j f Next, introducing a dimensionless time * Note that T here signifies a different dimensionless time from that used elsewhere 272 i 2 Trt T = (G 9) -(cbF) and a dimensionless potential scan rate aD ( cnF2 Tr (G.lO) so that .= at= rr, one obtains an integral equation for ewhich contains r, and a, and y, as parameters T (d@/dT) dT' (G 11) fo 4 To solve (11) numerically, we use the method of Acrivos and Chambr 69 (see Appendix F). The righthand side of (11) is thereby reduced to a summation and (11) to an algebraic equation co0[nT}1_tnT) - (O0 [nAT]) 1 e T) = L[O[i] (vTi - - O[n] + O[k]A k (G 12) where A - n-k = 2/V V'IZI Equation (12) is now solved for 6 at every, time-step, starting with O[l] = 1 The value of 80 at the previous step is used as the first trial in a Newton-Raphson root-finding iteration. The current f011ows • directly from (3): i0 (OY_ e T OY+e_TT) i = , (G 13) ,\LPH0 50 BTA0 50 GANNA0 5 N - : - iL • I't _ ••'••% -, IN •-- 0- 1 0 QE+ 00 N 0 OF+ 00 1 ' •••••% - N - et? 3 0 0U+ 00 N C 00U+0 -- LI L S IC C I Ci Ci C) I I LOG PHI XBL76-iO8 Fig. G.l., Current response to negative, potential scan of a deposition reaction characterized by a = =y 0.50. PHI = q, CURRENT = i/i 0 The scan -rate aD cb nF 2 r=—(. ) iT 1 5 0 varies from 10 (lowest Lurve) to 10 (top) by decades ALPH 4. QOF..O . QOE+QO F- a:Ui LOOE+QO a: (9 C) J -1.00E+OO OOE+PO OOE+OQ •.&. QOE+OO •-';. OE+OC • 0 C) C) C) 0 0 0 Cl 0 C) 0 0 C) 0 C) 0 C) Cl C) (1 C) 0 C) '1 0 -j LOG PHI XBL 716-1029 Fig. G.2. Current response to potential scanning Of a deposition reaction characterized by a = y 0.75, = 0.25. PHI p; CURRENT = i/i Scan rate increases by decades from r = (lowest curve) to 0 r=10 9 (top). 275 i.e., the left-hand or righthand side of (12) will give the current relative to I o The program THETA reproducedat the end of this ápendix calculates the concentration 0 and the current i/i, given input parameters RATE(r), ALPHA(ci), BETA(8), GM44A(y) and given an initial value of the potential increment rT. 201 meshpoints are used in the calculation. The initial 5 increnent of L4 = r -r is S x 10 . In the next time cycle the increment is multiplied by 10 and the first 20 meshpoints are selected from the last cycle. Seven decades were usually computed in this manner. The results of principal interest are the value of the current maximun and the potential at which it occurs. Since a systematic variation of the four input parameters is not very meaningful, a few combinations of • ct,8 andy have been selected and r has been varied systematically. Table G.i presents values of I = i/i and t at the maximum for = 0.50 (cL 1 = 0.50, y = 0.50), and for (cL = 0.75, = 0.25) with four different values of y; one of these values is 0.40 (cf. copper deposition electrode kinetics discussed in section 2.3). Figures G.1 and G.2 illustrate the current response to a scan rate varying over 15 decades. The scale is logarithmic in both coordinates. This makes it possible to distinguish the characteristic features of the diffusion limitation modified by electrode kinetics: a. All rates have a common short-time asymptote: = (a+)4 , ( G 14) this is the linear form of (2.29).. The slope of the asymptote is +1 They also have a common long-time asymptote, namely: = 1 , (GlS) / this is the response to a unit step in concentration at the surface. Expressed in units log(i/i ) and r (9), the long-term asymptote shifts a vertical distance log /10 for every tenfold increase in r. The slope of the asymptote is -112. The potential at which the maximum occurs is 0.860 at the lowest rates This indicates that diffusion control is dominant, since it tends to the value found when kinetics are left out of account (-0.8540R 0T/nF my, cf. section 4.7). The current peaks at low r follow the relation (4.28) which, expressed in I, r and T, reads I max = 1.0821v'i . (G.16) At high rates the maximum is shifted to more negative potentials by the electrode-kinetic constraint on the current. The envelope of the curves for high rates is the most simplified form of the Butler-volmer equation (2.29): I = e - e (G. 17) Note that the initial part of the envelope is linear, and also the part at very high (Tafel relation). The envelope for = 0.25 differs from that for = 0.50 by having an inflection, and by proportion- ately higher potentials for the same current. 0.4 279 e As can be expected from the behavior of the envelope, the current peaks•do not follow (16) anymore at rates above 10 butat high rates (> 10) they again tend to follow a / dependence: the pure-diffusion limitation (15) is then reached almost instantaneously. There remains, however, an "overshoot" of the current so that the maximum is not simply given by the intersection of the lines I = (G iSa) and I = exp(-k) . G.17a) f. The potential of the current maximum at high rates is strongly dependent on the value of (see Table G 1) Delahay 172 first pointed out the • importance of the shift in potential as an indicator of n for a particular reaction. The potential shift appears to be barely dependent on y, for the case = 0.25 (Table G.l). However, the current level at the peak is = dependent on and y, as shown in Figure G.3, where it is plotted relative to the pure-diffusion current (16), for = 0.25. If the value of is known, a study of current peaks in potential scanning should yield information about y. • Figure G 4 shows current peaks and plateaux obtained by potential scanning of copper deposition at a rotating disk electrode The plot is similar to that of Figure 4.13. The solid line represents (4.28), or equation 16, in terms of rotating disk variables One might now assume that the highest peak current of Figure G 4 lies already in that region of Figure G 3 where constant dependence on Vr is reached ft is by AL.PH '. Qf+Z (j OOE+O (!3 C) -J •-5. OQE+CO --.. OE+QO . •..' •... COE+O . I I I I I U16 ' LOG 'PHI XBL7ib-OZ7 Fig. G.S. The ratio of concentration overpôtential to surface overpotential in potential scanning at rates from r = (highest curve) to 2 = 10 9 (lowest), for a deposition reaction with cv. = = y = 0.50. 30f+OC ALPH !• 2. OO+QO I-4 4: £k. 00f+OQ 0 a. QQE+OO J. -5. OOF+OO -a. OOE+OO l. OOE+OQ .-1. 0 0 (1 Cl C) C) C) C) C) C) $ $ $ $ UI UI $ + $ UI UI ILl UI UI LII ILl UI C C) C) () C) C) ) C) C) C) 'I LI) 4 0 !i) LOG 'PHI XBL.716-033 Eig G 6 The ratio of concentration overpotential to surface overpotential in potential scanning at rates from r = (highest curve) to r = iü (lowest), for a deposition reaction with c = y = 0 75, = 0.25. 10 •282 approximation 2.5x10 3 ). The value of the current is, compared with the pure-diffusion current, 1/idiff 0.32 This indicates a very high value of y, in accordance with the findings of Chapter 4. • g. Figures G.5 and G.6 show the behavior of the ratio concentration to surface overpotential. The ratio is large for small rates and goes through a fairly flat maximum. For the higher rates a dependence on VF exists at small times. This follows from: • . i = 2cnFa /Dt/rr . (G.18) which is the short-term approximation of (4.26), and (14). At •long times the ratios for all scan rates converge. The approximate convergence ratios are: . •ot= 0.50 0.50 y = 0.50 = 0.50 0.75 0.25 . 0.75 0.25 • • . 0.75 0.25 0.50 • 0.33 • • • 0.75 • • 0.25 0.40 0.38. • 1• 0.75 • 0.25 0.30 • • 0.45 • These values are accurate to + 1%. For the same reason as indicated • • in Appendix F the accuracy of the numerical procedure, is limited with .respect • to the behavior at large time. This explains why the. values of 0 in Table G.1 • do not remain stable as r is increased but vary slightly. 283 Table.G.l. current maximum and corresponding potential for various scan rates. c1=050 =0.50 I=0.25 y=0.S0 =075 logr I 00 I 00 -5 3.417E-3 0.860 0.425 3.413E-3 0.860 0.426 -4 1 077E-2 0.875 0.424 1 073E-2 0.875 0.425 -3 3 368E-2 0.925 0.418 3 335E-2 0.925 0 423 -2 1.032E-1 1.05 0.411 1.002E-1 1.05 0.427 -1 3 000E-1 1.45 0.380 2 757E-1 1.50 0 413 0 8 138E-1 2.40 0.336 6 618E-1 2.65 0.412 1 2 205E0 4.25 0.278 1 579E0 6 70 0.297 2 6 453E0 6.60 0 239 4 559E0 12.0 0.227 3 1 988El 8.90 0.232 1 406E1 16 5 0.227 4 6.24SE1 11.5 0 199 4 411E1 21.0 0.231 5 1.973E2 13.5 0.231 1.391E2 25.5 0.237 6 6.227E2 :16.0 0.209 4.397E2 30.5 0.215 7 1 966E3 18.0 0.243 1 391E3 35.0 0.220 8 6 226E3 20.5 0.220 4 397E3 39.5 0.226 9 1 965E4 23.0 0.199 1 390E4 44 0 0.232 284 Table Gi. (continued) a=0.75 w0.25 c=0.75 =025 y=0.50 y=0.40 log r 00 -5 3.416E-3 0.860 0.425 3.417E-3. 0.860 0.425 -4 1.077E-2 0.875 0.424 1.078E-2 0.875 0.423 3.366E-2 0.925 0.418 3.377E-2 0.920 0.418 -2 1.029E-1 1.10 0.395 1.038E-1 1.10 0.390 • -1 2.930E-1 1.50 • 0.381 2.998E-1 1.50 0.369 0 • 7..298E-1 2.70 0.354 7.617E-1 2.70 0.330 1 1.753E0 6.30 0.261 1.838E0 6.10 0.248 • 2 5.012E0 11.0 0.219 5.234E0 11.0 0.186 3 1.540E1 • • 15.5 0.218 1.609E1 15.5 0.185 4 4.832E1 20.5 0.190 5.042E1 • 20.0 0.190 S. 5 1.525E2 25.0 0.196 1.590E2 • 24.5 0.197 • 6 • • 4.818E2 29.5 0.203 5.021E2 29.5 0.169 7 1.523E3 34.0 0.210 1.589E3 34.0 0.176 285 • Table.G.1. (continued) • . • • • • c=0.75 =0.25 AL • • y= 0.30 • . • logr 1 -S 3.419E-3 .0.860 • 0.425 • • • 1.079E 7 2 0.875 0.423 . . 3.386E-2 • 0.920 0.417 . . . -2 . . 1.047E-1 1.05 .0403 • 3 .06 7- 1 • 1.50 0.356 0 7 967E-1 2.70 0.305 1 1937E0 595 0227 5.493E0 10.5 . 0.187 . • . . 1.685E1. 150 0.18 . . . . 4 5.277E2 20.0 0.153 . . . . 5 . 1.666E2 .24.5 . 0.160 . . . . . 6 5.266E2 29.0 0.167 7 1.665E3 33.5 • 0.175 286 PROGR.v4 T-IETA (INPJT,OJTL'JT) 1) I::NS r JN TI ( 20 1) • ( 2 0 1) •3 2J 1) ,c'is C ..i 1) ,E T A A201) ,AT C 201. 1 102 FO.Ar( l')x,14,1lX,F').3 9 X9 F8.59X,11.5,X9H.6) 103 FORM'\T(19a,l5x,4r1 NZ ,7X ,1H PHI ,9H RATIO 912H TIT 1. 9144 C'JRRFNT 1i4 F0'1AT(1.)X,4HNZT= , I4 , 4H,DZ=,F1.5,6:1,SIGN=,F:.).3,6r1,RATE,F12.),1 l'1S TI'IE= ,16) 1 4,5 13.MAT(l-)X, 7H ALP-IA,F7.3,7H, BETA=,F7.3,7H,GA..,F l) 1)6 FO,R4AT(23X,F7.3, X,F3.59x,11.5,X9E13.6) 117 F1RAT(li5X,Hr)Z= ,F12.51 108 FORt'AT(/1 )X,4r-1 11,r)X2=9EI3.5,5H9.u:1z9F6.593H93=,F3.5,3H,N=9,I4,) ill CORM A, T 149 E12.39F3.0 9 12.59 14) 112 FOR'4AT(6F.2) 115 FoR'.AT(r?) 125 FORTt115X,5HRT= 9E3.l) 126 FORAT(115X, 127 FORAT(l1'X, 7)FVi = ,E.2) 199 FOR"AT(IHO,flx, 7H PHI 9 9H RATIO 912H T H E T i: 914H ''JRRENT I 999 113.6939, 7= , E13. 6, 3H9I9 14/) 1110 FOAl (19O,NO CONVERGENCE AT*,F6.3) 1111 FORMAT( 191 1 1112 FOR'IAT (190) '4M0 DFV = • 00000 1. C MMAX= NJ'3ER OF CASES WITH DIFFERENT RATES REAP) 1159W'4X REAr i1i,rZT,r)Z,.S1GN,RT,: -) Fc REAl 112,AL 2 HA,RFTA.,GA.1MA 70 TH(l)=l. S=TH(1) $EX=).5 DEVM=DEVI 5 F T AA 1 0 S CUR(1)=0. RAT(1)O. I F (3 T A X 50 TO 119 RATF=RATE*11. .¶ D7=DZ/1•'.NDEC JAEGIN=2 • KKMAX=(NZT-1)/1) + 1 00 51 N?=1,NT I ANZ=N7 A(NZ).ANEx -(ANZ+l.)**EX _(ANZ1.)**Ex 6 (NZ)=A7':EX - (ANZ-l.)x- EX °RINT 1111 °RINT 114 NZT , 07., SIGNS RATE NDEC • PRI',IT 115 9ALPi1A,ETA,GA'i1A PRI!IT 1I N TW=7 1 CALr, SJ1(TH(J)UA(K)) 00 KN7-J+1, 40 SU'4 = S )i + T4(J 287 4 NJ= "I Z - 1 56 N = 1,2) IF.(S.'T.o. ') TO 129. 12 ,3 FYM.'\T(1')5X,1lHS rj r IT-,t4) IF 1JLA5.').1) DRINT 128'N JFLAG=l cuR I NZ — l) *FXP( 3FT A*F TA) ) *( GAMi1 TA) 129 Xl = EX?(ALPHA*ETA) r)x I Xl*.IGA.ltIA._ALPHA),s • )(2=. 3TA) cXP(—'iETA*ETA) X3= T'.I.*(TH(1)*.'3(NJ) - S + S1J'1)I oz:*Ex C'J(NZ)=X3 0T= S - I( (X1—X2)#X3)/(.)x1_r)x2 _T4IrZ*Ex) IF(CIJ;(NZ).L.j.UQ1) PRINT 1J8 , X1 , X2 , X3,DX1,DX2,SUi,6 9N • IF(.N.Lt.e3) SO TO 56 0EV = (5 - r)TH)/rTLf 53 I (ARS(V) .L. r)EV'1) (O TO 60 56 S 0TH RIT tOlO,ETA 60 TH(!Z) = 0TH - S + SUM)/DZ**Ex 62 ETAC=A'5(AL0GTH(NZ) RAT(NZ)—TAC/ETAA(NZ) PRINT 1029NZ9 ETA ,RAT(NZ),TH(NZ),CUR(NZ) 80 CONTINrJE 117 00.11 K1KKMAX JK*i0-9 TH(K)=TH(J) RAT ( K) =RAT( J) CUR (K) =CJR ( J) ET(<)=ETAA( J. I CONTJN') DZ=l0.*DZ PRINT 1111 PRINT 125,ATE MW4=.'1+ 1. PRINT 126,W4M PRINT 107,02 PRINT 127, !')EV"l PRjjT 199 PRINT 1063(ETAA(N 7 )5RAT(NZ),TJ(NZ) ,CUR(NZ) ,NZ=1,KKMAX) PRINT 1112 PRINT 103 • Jr3EGINKK44x+l . • 3 CONTINUE PR1MT liii DRINT 1112 GO 1') 73 119 COTrIN'JE • • 11 1. CONTINuE STOP , END 20 +20]4 'V)5 r7_•4•00"y)-,_6 07 0.75 0.25 0I' 288 NOMENC LATURE a thermal diffusivity [cm2 /sec] a dimensional coefficient (1.35) [cm 2 ] a radius of model sphere (Ch.. 3) [cm] a dimensionless potential scan rate (Ch 4) a' potential scan rate [mv/sec] a activity of species i [mole/cm3 ] ar critical electrode-diaphragm separation (1.25) [cm] 2 A area [cm] b dimensional coefficient (1 35) [CM71 concentration . . [mole/cm3 ] C constant in mass-transfer correlations (Ch 1) D diffusivity . [cm2 /sec] e electron E electrode potential [v] EP standard electrode potential [v] LE applied cell voltage . [v] . H f frequency (Ch. 1) . [sec1] f . fraction of limiting current (2.31, App. A) activity coefficient of species i F Faraday's constant [96,494 C] g . acceleration of gravity . [cm/sec 2] G mass flux [g/cm2 secj h distance between electrodes (Ch 2) [cm] h mesh size (App B) 289 i current density fma/cm2] • limiting diffusion current [ma/cm 2 ] iL limiting current [ma/cm2] 10 exchange current density [ma/cm2] 'urn limiting current [ma/cm2] I apparent ionic strength [mole/Z] true ionic strength [mole/fl effective current on rough surface (3.14) [ma] current on smooth surface (3.14) [ma] I dimensionless current density (App. F, G) k mass-transfer coefficient [cm/see] k rate-èonstant electrochemical reaction (2.40) [cm/sec] K dissociation constant (thermodynanic) [mole/i] K' • dissociation constant (stoichiometric) [mole/i] L length or height [cm] M. species participating in charge-transfer reaction (2.1) n number of electrons transferred in charge-transfer reaction (2.1) i'i dimensionless frequency (Ch 1) 2 N . flux • •. [mole/cm sec] AP pressure drop [dyne/cm2] Q volume flow rate [cm3/sec] r radius (r inner, r outer radius) [cm] r ratio of supporting to total electrolyte (2 3) r dimensionless potential scan rate (G 10) 290 R ohmic resistance of electrochemical cell (Ch. 2) [c] • R diffusion paranieter for redox reaction (4.33;. App. F) • R0 gas-constant [8.3143 VC/mole ° K] • • s area per unit volume in packed bed (Ch. 1) [cm] s • flux increase rate (4.16) • . [cm/sec] • s. number of ions or molecules of species i participating in charge-transfer reaction (2.1) S velocity gradient at mass-transfer interface [sec] t . time . . [sec] t. transference number of species I T temperature [ ° K] I dimensionless time in Appendix.B (B.23a) mobility of ionic species i • . [cm2mole/Jsec] . v velocity . [cm/sec] •• . . x longitudinal distance from leading edge of electrode [cm] or interface . . . . y distance perpend.icular to electrode or interface [cm] • z valence of ionic species . . Z charge transfer number defined by (2.29a,b) . . . 291 GREEJ( 'SYMBOLS exponential coefficient (charge-transfer coefficient) in (2.29) ci. degree of dissociation of HSO ion (App. C) densification coefficient due to concentration gradient of species 1 [cm3/mole] exponential coefficient (charge-transfer coefficient) in (2.29) curvature parameter in free convection (Fig. 1.2) y exponential coefficient in (2.30) T(4/3) = 0.893 thickness of boundary-layer or film [cm] C porosity, field strength [v/cm] C densification coefficient due to temperature gradient [OK_ l i ri dimensionless distance variable (App B,E) Ti overpotential [v] concentration overpotential [v] TI surface overpotential S [v] o dimensionless concentration (c/cb) 0 degree of dissociation of CuSO4 , in App C 292 Subscripts . . . . avg averaged . . b in the bulk . B pertaining to bisulfate dissociation C pertaining to copper sulfate association d with respect to equivalent diameter (Ch. 1) h pertaining to heat transfer i. pertaining to species i m modified o at the electrode or interface . ox pertaining to oxidized species p with respect to particle dimensions red pertaining to reduôed species R pertaining to reacting species s pertaining to the molecule (salt) as a. whole ss corresponding to ste.ady state + with respect to wall-parameters (Ch. 1) pertaining to positive or negative ion CO corresponding to steady state CO in the bulk (Figures 2.3, 2.8, 2.11) Superscripts . . o at the electrode (Fig. 2.3) - . averaged . corre.sponding to uniform flux condition CO in the bulk (Fig. 2.3) 293 M dynanic viscosity [cp] kinematic viscosity [cst] p density [g/ml] -1/3 T dimensionless time wt Sc (Ch. 3,4) T. dimensionless time 0. 307 uit Sc 1 " 3 (App. B,E) T shear-stress at the wall [dyne/cm2 ] auxiliary function in App. B parameter in laminar transfer equation for • • • annular ducts (1.6, fig. 1.1) dimensionless potential (Ch. 4, App. A, F, G) potential • [v] • rotation rate • [radians/sec] • 4 294 Dimensionless Numbers . . . . 3 Gr Grashof number (uniform concentration) gthcx 2 . : V Gr Grashof number (uniform flux) xNX VD • 3D Chilton-Colburn parameter —Sc •. • ViLZF • . . • • J • current distribution parameter °R T • • • • • • • • •• • ba • • N • •, current distribution parameter • • • (2.57) • • N' current distribution parameter. 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